TEST BANK for Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences 14th Ed by StudyGuide

CHAPTER 1

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the inequality and graph. Express your answer in interval notation. 1) 4x - 4 > 3x - 1

A) [3, ∞) -3 -2 -1

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

B) (3, ∞) -3 -2 -1

C) (-5, ∞)

D) (-∞, 3] -3 -2 -1

Provide an appropriate response. 2) Write the equation of a line that passes through (-1, 4) and (5, -1). Write the final answer in the form Ax + By = C where A, B, and C are integers with no common divisors (other than ±1) and A > 0. A) 5x + 6y = 19 B) 5x + 6y = -19 C) 5x - 6y = 19 D) -5x + 6y = 19 3) Find the slope of the line 3x + 4y = 11. 3 A) B) - 3 4 4

3) C) - 4 3

Graph the linear equation and determine its slope, if it exists. 4) 2x - 5y = 20 10

8 6 4 2 -10 -8 -6 -4 -2 -2

8 10 x

-4 -6 -8 -10

D) 0

A) slope = 2 5

B) slope = - 2 5 y

-10 -8 -6 -4 -2 -2

8 10 x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

C) slope = - 2 5 y

-10 -8 -6 -4 -2 -2

8 10 x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

Solve the inequality and graph. Express your answer in interval notation. 5) -29 ≤ -4x - 1 ≤ -13

A) (3, 7) 0

9 10 11

B) [-7, -3] -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

C) [3, 7] -1

8 10 x

D) slope = 2 5 10

-1

9 10 11

D) (-7, -3) -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

Determine whether the slope of the line is positive, negative, zero, or undefined. 6)

y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

10 x

-4 -6 -8 -10

A) zero

B) undefined

C) negative

Solve the inequality and graph. Express your answer in interval notation. 7) -3(3x - 2) < -12x - 21

A) (-∞, -9] -12

-11

-10

-9

-8

-7

-6

-11

-10

-9

-8

-7

-6

-11

-10

-9

-8

-7

-6

-11

-10

-9

-8

-7

-6

B) (-9, ∞) -12

C) [-9, ∞) -12

D) (-∞, -9) -12

D) positive

8) 24x + 8 > 4(5x + 7)

A) [5, ∞) 2

B) (5, ∞) 2

C) (-∞, 5) 2

D) (-∞, 5] 2

Use the REGRESSION feature on a graphing calculator. 9) A study was conducted to compare the average time spent in the lab each week versus course grade for computer students. The results are recorded in the table below. Hours in lab 10 Grade (percent) 96

11 51

16 9 7 15 16 10 62 58 89 81 46 51

Use linear regression to find a linear function that predicts a student's course grade as a function of the number of hours spent in lab. A) y = 0.930 + 44.3x B) y = 44.3 + 0.930x C) y = 1.86 + 88.6x D) y = 88.6 - 1.86x Use the graph to find the average rate of change. 10)

10)

y 10

-10

-5

10 x

-5

-10

A) -

1 2

B) - 2

C) 2

1 2

Solve the problem. Express your answer as an integer or simplified fraction. 5x - 7 = 7x + 3 11) 5 2 A) - 1 25

1 45

C) - 29 25

11) D)

29 45

Write the slope-intercept equation (y = mx + b) for a line with the given characteristics. 12) m = - 4, y-intercept (0, -7) A) y = - 4x B) y = - 4x - 7 C) 4x + y = - 7 D) y = - 7x - 4 Find the slope of the line containing the given points. 13) (-5, 2) and (0, 2) 5 A) B) 0 2

12)

13) C) - 5 2

D) Undefined

Provide an appropriate response. 14) Use the graph to find the slope-intercept form of the equation of the line.

14)

y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

10 x

-4 -6 -8 -10

A) y = x + 3

B) y = -x + 3

C) y = 3x

D) y = x - 3

15) Find the standard form of the equation of the line with slope of - 2 and passing through (4, 4). 7 A) 2x + 7y = 36

B) 2x + 7y = - 36

C) 2x - 7y = 36

D) 7x + 2y = - 36

Find the slope and y intercept of the graph of the equation. 3 16) y = 5 x 2 2 A) Slope =

16)

3 ; y intercept = 5 2 2

3 B) Slope = 5 ; y intercept = 2 2

3 ; y intercept = 5 2 2

3 D) Slope = 5 ; y intercept = 2 2

C) Slope = -

15)

Solve the problem. Express your answer as an integer or simplified fraction. 17) Solve: x - 2 - x - 3 = 3 - x - 3 3 6 2 A) 2

B) 3

C) - 3

Write an equation of the line with the indicated slope and y intercept. 18) Slope = 1; y intercept = -5 A) y = -5x - 1 B) y = x - 5 C) y = -x - 5

17) D) - 2

18) D) y = -5x + 1

Use the REGRESSION feature on a graphing calculator. 19) For some reason the quality of production decreased as the year progressed at a flash drive manufacturing plant. The following data represent the percentage of defective flash drives produced at the plant in the corresponding month of the year. Month, x % defective, y

2 1.3

3 1.6

5 2.0

7 2.4

8 2.6

19)

9 12 2.8 3.1

Use the regression equation with values rounded to four decimals to predict the percentage of defective drives in month 6, June. A) 2.15% B) 2.3% C) 2.20% D) 2.0% Write an equation of the line with the indicated slope and y intercept. 7 20) Slope = 5 ; y intercept = 2 2 A) y = -

7 x+ 5 2 2

21) Slope = 2, y intercept = -4 A) y = -2x - 4

20)

7 B) y = 5 x + 2 2

7 C) y = 5 x 2 2

D) y =

7 x- 5 2 2

B) y = 2x - 4

C) y = 4x - 2

D) y = 4x + 2

21)

Determine whether the slope of the line is positive, negative, zero, or undefined. 22)

22)

y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

10 x

-4 -6 -8 -10

A) negative

B) zero

C) positive

D) undefined

Write an equation of the line with the indicated slope and y intercept. 23) Slope = - 1 ; y intercept = -5 2 A) y = -5x + 1 2

B) y = x - 5 2

C) y = -5x - 1 2

23) D) y = - x - 5 2

Solve the problem. 24) The cost of manufacturing a computer part is related to the quantity produced, x, during a production run. When 100 parts are produced, the cost is $300. When 600 parts are produced, the cost is $ 4800. Find an equation of the line relating quantity produced to cost. Write the final answer in the form C = mx + b. A) C = 9x B) C = 9x - 600 C) C = 600x + 9 D) C = 9x + 600 Provide an appropriate response. 25) Given two points (x1, y 1) and (x2, y 2), the ratio of the change in y to the change in x is called. A) break-even point C) equilibrium point

Solve the problem. 27) The cost for labor associated with fixing a washing machine is computed as follows: There is a fixed charge of $25 for the repairman to come to the house, to which a charge of $20 per hour is added. Find an equation that can be used to determine the labor cost, C, of a repair that takes x hours. Write the final answer in the form C = mx + b. A) C = -20x + 25 B) C = 45x C) C = 25x + 20 D) C = 20x + 25 28) Find the Celsius temperature (to the nearest degree) when Fahrenheit temperature is 50° by solving the equation 50 = 9 C + 32, where F is the Fahrenheit temperature (in degrees) and C is 5 B) 122°C

C) 10°C

B) Slope = 5; y intercept = 1 2

C) Slope = - 1 ; y intercept = -5 2

D) Slope = - 1 ; y intercept = 5 2

Solve the problem. Express your answer as an integer or simplified fraction. x -4= x -3 30) 6 3 C) - 14 7

27)

28)

29)

A) Slope = 5; y intercept = - 1 2

B) - 6

26)

D) 96°C

Find the slope and y intercept of the graph of the equation. 29) y = - x + 5 2

A) - 2

25)

B) x-intercept D) slope

Write the slope-intercept equation (y = mx + b) for a line with the given characteristics. 26) m = 3, passing through (1, -2) A) y = 3x B) y = 5x - 3 C) y = 3x - 5 D) y - 5 = 3x

the Celsius temperature. A) 24°C

24)

30) D) 14

Provide an appropriate response. 31) Find the standard form of the equation of the line passing through the two points. (2, - 6) and (- 9, 6) A) - 8x + 15y = - 18 B) 12x + 11y = - 42 C) 8x - 15y = - 18 D) - 12x + 11y = - 42

31)

Use the REGRESSION feature on a graphing calculator. 32) Efficiency experts rate employees according to job performance and attitude. The results for several randomly selected employees are given below.

32)

Attitude, x 59 63 65 69 58 77 76 69 70 64 Performance, y 72 67 78 82 75 87 92 83 87 78 Find the regression line which can be used to predict performance rating if attitude rating is known. A) y = 92.3 - 0.669x B) y = -47.3 + 2.02x C) y = 11.7 + 1.02x D) y = 2.81 + 1.35x Solve the problem. Express your answer as an integer or simplified fraction. 33) -2(2x + 5) - 4 = -2(x + 2) + 4x 5 1 A) B) C) - 5 3 6 Find the slope of the line containing the given points. 34) (6, 1) and (6, - 4) A) 0 B) - 1 4

33) D)

34) C) - 4

D) Undefined

Use the graph to find the average rate of change. 35)

35)

y 10

-10

-5

10 x

-5

-10

A) 1

5 6

B) -4

C) -1

D) 4

Use the REGRESSION feature on a graphing calculator. 36) The paired data below consists of the temperature on randomly chosen days and the amount of a certain kind of plant grew (in millimeters). Temp, x Growth, y

62 36

76 50 39 50

51 13

71 33

46 33

36)

51 44 79 17 6 16

Find the linear function that predicts a plant's growth as a function of the temperature. Round your answer to two decimal places. A) y = 14..57x + 0.21 C) y = - 9.19x3 + 0.11x2 - 2.90x + 6.54

B) y = 0.21x + 14.57 D) y = - 0.06 x2 + 7.20x - 191.23

Solve the formula for the specified variable. 37) Solve: D = 4 (mx - mb) for m 5 A) m =

4D 5(x + b)

B) m =

37) 5D 4(x + b)

C) m =

Find the slope and y intercept of the graph of the equation. 38) y = x + 3 A) Slope = 3; y intercept = 1 C) Slope = 0; y intercept = -3

5D 4(x - b)

4D 5(x - b)

D) m =

38) B) Slope = 3; y intercept = -1 D) Slope = 1; y intercept = 3

Solve the problem. Express your answer as an integer or simplified fraction. 39) 7x - (5x - 1) = 2 1 1 A) - 1 B) C) 2 12 2

39) D) - 1 12

Write an equation of the line with the indicated slope and y intercept. 3 26 40) Slope = - ; y intercept = 5 5 A) y = -

3 26 x+ 5 5

B) y = -

3 26 x5 5

C) y = -

40) 5 26 x+ 3 5

D) y =

3 16 x+ 5 5

Provide an appropriate response. 41) Find the line passing through the two points. Write the equation in standard form. (10, 9) and (10, 1) A) x + y = 11 B) x = 10 C) x + y = 19 D) y = 9 Find the slope of the line containing the given points. 42) (5, -3); (-8, 2) 13 13 A) B) 5 5

41)

42) 5 C) 13

5 D) 13

Solve the problem. 43) At a local grocery store the demand for ground beef is approximately 50 pounds per week when the price per pound is $4, but is only 40 pounds per week when the price rises to $5.50 per pound. Assuming a linear relationship between the demand x and the price per pound p, express the price as a function of demand. Use this model to predict the demand if the price rises to $5.80 per pound. A) p = 11.5x + -0.15; 40 pounds B) p = - 0.15x + 11.5; 38 pounds C) p = - 0.15x - 11.5; 40 pounds D) p = 0.15x + 11.5; 38 pounds 44) Assume that the price per unit d of a certain item to the consumer is given by the equation d = 35 - .10x, where x is the number of units in demand. The price per unit from the supplier is given by the equation s = .2x + 20, where x is the number of units supplied. Find the equilibrium price and the equilibrium quantity. A) equilibrium price: $30 per unit; equilibrium quantity: 50 units B) equilibrium price: $20 per unit; equilibrium quantity: 50 units C) equilibrium price: $50 per unit; equilibrium quantity: 30 units D) equilibrium price: $35 per unit; equilibrium quantity: 50 units Graph the linear equation and determine its slope, if it exists. 45) 3x + 5y = 11 y

8 6 4 2 2

8 10

-4 -6 -8 -10

A) slope: 3 4

B) slope: - 3 4 y

-10 -8 -6 -4 -2 -2

8 10

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

44)

45)

-10 -8 -6 -4 -2 -2

43)

8 10

C) slope: 3 4

D) slope: - 3 4 y

-10 -8 -6 -4 -2 -2

8 10

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

8 10

Solve the problem. Express your answer as an integer or simplified fraction. x - 5 = x+6 46) 16 8 8 A) - 16

B) - 17

C) - 22

Solve the formula for the specified variable. 47) 7x + 10y = 19 for y A) y = - 7 x + 19 10 10

46) D) - 11

47) B) - 7x - 10y = -19

C) y = 7 x + 19 10 10

D) y = 7x - 19

Use the REGRESSION feature on a graphing calculator. 48) The use of bottled water in the United States has shown a steady increase in recent years. The table shows the annual per capita consumption for the years 1995 - 2001. Year 1995 1996 1997 1998 1999 2000 2001 Gallons/person 4.4 5.1 5.7 6.4 7.3 8.0 10.2 With x being the years since 1995, find the linear function that represents this data. Round your answer to two decimal places. A) y = 0.1x2 + 0.29x + 4.57 B) y = 4.07x + 0.89 D) y = 0.04x3 - 0.23x2 + 1.01x + 4.35

C) y = 0.89x + 4.07

48)

Provide an appropriate response. 49) Write the equation of the line in the following graph.

49)

y 3 2 1 -6 -5 -4 -3 -2 -1 -1

1 2 3 4 5 6x

-2 -3

A) f(x) = 1 x - 1 3

B) f(x) = 1 x + 1 3

C) f(x) = - 1 x + 1 3

D) f(x) = - 1 x - 1 3

Solve the formula for the specified variable. 50) F = 9 C + 32 for C 5 A) C = 9 (F - 32) 5

B) C =

50) 5 F - 32

C) C = 5 (F - 32) 9

D) C = F - 32 9

Determine whether the slope of the line is positive, negative, zero, or undefined. 51)

51)

y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

10 x

-4 -6 -8 -10

A) zero

B) undefined

C) positive

Provide an appropriate response.

D) negative

52) Graph the linear function defined by f(x) = 2 x + 2 and indicate the slope and intercepts. 3 5 4 3 2 1 -5 -4 -3 -2 -1 -1

1 2 3 4 5 x

-2 -3 -4 -5

A) x-intercept = -2; y-intercept = 3; slope 2 3 5 4 3 2 1

1 2 3 4 5 x

-5 -4 -3 -2 -1 -1 -2 -3 -4 -5

B) x-intercept = 3; y-intercept = -2; slope 2 3 5 4 3 2 1 -5 -4 -3 -2 -1 -1

1 2 3 4 5 x

-2 -3 -4 -5

52)

C) x-intercept = -3; y-intercept = 2; slope 2 3 5 4 3 2 1

1 2 3 4 5 x

-5 -4 -3 -2 -1 -1 -2 -3 -4 -5

D) x-intercept = 2; y-intercept = -3; slope 2 3 5 4 3 2 1 -5 -4 -3 -2 -1 -1

1 2 3 4 5 x

-2 -3 -4 -5

Graph the equation. 53) 42 + 6y = 0

53) y

10 5

-10

-5

10 x

5 -5 -10

B) 10

-10

-5

10 x

-10

-5

-10

10 x

D) 10

-10

-5

10 x

-10

-5

-10

Solve the problem. 54) Suppose the sales of a particular brand of MP3 player satisfy the relationship S = 200x + 3800 , where S represents the number of sales in year x, with x = 0 corresponding to 2002 . Find the number of sales in 2005. A) 4200 B) 12,600 C) 6400 D) 4400 Provide an appropriate response. 55) Find the line passing through the two points. Write the equation in standard form. (-3, 6) and (6, 6) A) -2x - y 0 B) -x - 2y = 0 C) x = -2 D) y = 6

54)

55)

Determine whether the slope of the line is positive, negative, zero, or undefined. 56)

56)

y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

10 x

-4 -6 -8 -10

A) negative

B) positive

C) zero

D) undefined

Find the slope and y intercept of the graph of the equation. 4 18 57) y = - x + 5 5 A) Slope =

57)

4 8 ; y intercept = 5 5

B) Slope =

5 8 ; y intercept = 4 5

4 18 ; y intercept = 5 5

D) Slope =

4 18 ; y intercept = 5 5

C) Slope = -

Solve the problem. 58) A piece of equipment was purchased by a company for $10,000 and is assumed to have a salvage value of $3,000 in 10 years. If its value is depreciated linearly from $10,000 to $3,000, find a linear equation in the form V = mt + b, t time in years, that will give the salvage value at any time t, 0 ≤ t ≤ 10. A) T = - 700V + 10,000 B) V = - 700t - 10,000 C) V = - 700t + 10,000 D) V = 700t + 10,000 Solve the problem. Express your answer as an integer or simplified fraction. 1 1 59) - (x - 18 ) - (x - 8 ) = x - 7 6 8 A)

120 31

264 31

216 31

58)

59) D)

72 31

Provide an appropriate response. 60) Use the graph to find the slope, x-intercept and y-intercept of the line.

60)

y 10 8 6 4 2 -10 -8 -6 -4 -2 -2

10 x

-4 -6 -8 -10

A) slope = -1 x-intercept = (7, 0) y-intercept = (0, -7) C) slope = 1 x-intercept = (0, 7) y-intercept = (-7, 0)

B) slope = 1 x-intercept = (7, 0) y-intercept = (0, -7) D) slope = - 1 x-intercept = (-7, 0) y-intercept = (0, 7)

Solve the inequality and graph. Express your answer in interval notation. 61) -4(-2 - x) < 6x + 19 - 11 - 2x

A) (-∞, 8)

61)

B) ∅

-5 -4 -3 -2 -1

C) (-∞, 0)

-5 -4 -3 -2 -1

D) (-∞, ∞)

-5 -4 -3 -2 -1

Solve the problem. 62) The mathematical model C = 600 x + 30,000 represents the cost in dollars a company has in manufacturing x items during a month. Using this model, how much does it cost to produce 600 items? A) $0.08 B) $390,000 C) $360,000 D) $50.00 Find the slope and y intercept of the graph of the equation. 63) y = -4x + 6 A) Slope = 6, y intercept = -4 C) Slope = -6, y intercept = -4 17

62)

63) B) Slope = 4, y intercept = -6 D) Slope = -4, y intercept = 6

Use the REGRESSION feature on a graphing calculator. 64) In the table below, x represents the number of years since 2000 and y represents sales (in thousands of dollars) of a clothing company. Use the regression equation to estimate sales in the year 2006. Round to the nearest thousand dollars.

64)

Year x 1 2 3 4 5 Sales y 84 76 39 30 26 A) $20,000

B) $2,000

C) $14,000

D) $8,000

Solve the problem. 65) A small company that makes hand-sewn leather shoes has fixed costs of $320 a day, and total costs of $1200 per day at an output of 20 pairs of shoes per day. Assume that total cost C is linearly related to output x. Find an equation of the line relating output to cost. Write the final answer in the form C = mx + b. A) C = 60x + 320 B) C = 44x + 1520 C) C = 44x + 320 D) C = 60x + 1520 66) You have $50,000 and wish to invest part at 10% and the rest at 6%. How much should be invested at each rate to produce the same return as if it all had been invested at 9%? A) $37,000 at 10%, $13,000 at 6% B) $37,500 at 10%, $12,500 at 6% C) $37,500 at 6%, $12,500 at 10% D) $37,000 at 6%, $13,000 at 10% Solve the formula for the specified variable. 67) S = 2!rh + 2!r2 for h A) h = S - 2!r 2!r

65)

66)

67)

B) h = S - r

C) h = 2!(S - r)

D) h = S - 1 2!r

Provide an appropriate response. 68) Write the equation of a line that passes through (3, 9) and (0, -7). Write the final answer in the form Ax + By = C where A, B, and C are integers with no common divisors (other than ±1) and A > 0. A) 3x - 16y = 21 B) -16x + 3y = 21 C) 16x - 3y = -21 D) 16x - 3y = 21 Write an equation of the line with the indicated slope and y intercept. 69) Slope = -3, y intercept = 5 A) y = -3x - 5 B) y = 5x - 3 C) y = -3x + 5

69) D) y = 3x + 5

Solve the problem. 70) Using a phone card to make a long distance call costs a flat fee of $0.85 plus per $0.19 minute starting with the first minute. Find the total cost of a phone call which lasts 8 minutes. A) $6.00 B) $2.37 C) $1.52 D) $8.16 Find the slope and y intercept of the graph of the equation. 71) y = 2x - 6 A) Slope = 2, y intercept = -6 C) Slope = 2, y intercept = 6

68)

70)

71) B) Slope = 6, y intercept = 2 D) Slope = -6, y intercept = 2

Answer Key Testname: CHAP 01_14E

1) B 2) A 3) B 4) D 5) C 6) C 7) D 8) B 9) D 10) A 11) C 12) B 13) B 14) B 15) A 16) D 17) D 18) B 19) A 20) C 21) B 22) D 23) D 24) B 25) D 26) C 27) D 28) C 29) D 30) B 31) B 32) C 33) A 34) D 35) A 36) B 37) C 38) D 39) B 40) A 41) B 42) C 43) B 44) A 45) B 46) C 47) A 48) C 49) D 19

Answer Key Testname: CHAP 01_14E

50) C 51) A 52) C 53) D 54) D 55) D 56) B 57) C 58) C 59) B 60) B 61) B 62) B 63) D 64) B 65) C 66) B 67) A 68) D 69) C 70) B 71) A

CHAPTER 2

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) For f(t) = 3t + 2 and g(t) = 2 - t2, find 4f(3) - g(-3) + g(0).

2) The following graph represents the result of applying a sequence of transformations to the graph of a basic function. Identify the basic function and describe the transformation(s). Write the equation for the given graph.

y 9 8 7 6 5 4 3 2 1 -5 -4 -3 -2 -1

1 2 3 4 5 x

3) For f(t) = 3 - 5t, find f(a + h) - f(a) . h

4) Find the vertex and the maximum or minimum of the quadratic function f(x) = -x2 - 4x + 5 by first writing f in standard form. State the range of f and find the

intercepts of f . 5) If f(x) =

x- 3 x2

if x < 2

, what is the definition of g(x), the function whose graph is if x ≥ 2 obtained by shifting f(x)'s graph right 5 units and down 1 unit?

Solve the problem. 6) In the table below, the amount of the U.S. minimum wage is listed for selected years. U.S. Minimum Wage Year 1961 1967 1974 1980 1981 1990 1991 1996 1997 Wage $1.15 $1.40 $2.00 $3.10 $3.35 $3.80 $4.25 $4.75 $5.15 Find an exponential regression model of the form y = a ∙ b x, where y represents the U.S. minimum wage x years after 1960. Round a and b to four decimal places. According to this model, what will the minimum wage be in 2005? In 2010?

7) The financial department of a company that produces digital cameras arrived at the following price -demand function and the corresponding revenue function:

p(x) = 95.4 - 6x price-demand R(x) = x ∙ p(x) = x(95.4 - 6x) revenue function The function p(x) is the wholesale price per camera at which x million cameras can be sold and R(x) is the corresponding revenue (in million dollars). Both functions have domain 1 ≤ x ≤ 15. They also found the cost function to be C(x) = 150 + 15.1x (in million dollars) for manufacturing and selling x cameras. Find the profit function and determine the approximate number of cameras, rounded to the nearest hundredths, that should be sold for maximum profit. Provide an appropriate response. 8) Graph f(x) = -x2 - x + 6 and indicate the maximum or minimum value of f(x), whichever

exists. 8

6 4 2 -8 -6 -4 -2 -2

-4 -6 -8

Use the REGRESSION feature on a graphing calculator. 9) A particular bacterium is found to have a doubling time of 20 minutes. If a laboratory culture begins with a population of 300 of this bacteria and there is no change in the growth rate, how many bacteria will be present in 55 minutes? Use six decimal places in the interim calculation for the growth rate. Solve the problem. 10) The financial department of a company that manufactures portable MP3 players arrived at the following daily cost equation for manufacturing x MP3 players per day: C(x) = 1500 + 105x + x2. The average cost per unit at a production level of players per day is C(x) = C(x) . x (A) Find the rational function C. (B) Graph the average cost function on a graphing utility for 10 ≤ x ≤ 200. (C) Use the appropriate command on a graphing utility to find the daily production level (to the nearest integer) at which the average cost per player is a minimum. What is the minimum average cost (to the nearest cent)?

10)

Provide an appropriate response. 3 11) If g(x) = -4x2 + x - 9, find g(-2), g(1), and g . 2

11)

12) The following graph represents the result of applying a sequence of transformations to the graph of a basic function. Identify the basic function and describe the transformation(s). Write the equation for the given graph. 2 1 -14-12-10 -8 -6 -4 -2 -1

12)

2 4 6 x

-2 -3 -4 -5 -6 -7 -8 -9 -10

13) Let T be the set of teachers at a high school and let S be the set of students enrolled at that school. Determine which of the following correspondences define a function. Explain. (A) A student corresponds to the teacher if the student is enrolled in the teacher's class. (B) A student corresponds to every teacher of the school.

13)

14) Only one of the following functions has domain which is not equal to all real numbers. State which function and state its domain. (A) h(x) = 4x2 - 3x - 5 (B) f(x) = 2x (C) g(x) = x + 7 48 - x 2

14)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert to a logarithmic equation. 15) e t = 7 A) log 7 e = t

15) B) log 7 t = e

C) ln 7 = t

D) ln t = 7

Find the function value. 2 16) f(x) = x + 8 ; f(5) x3 - 2x A)

33 125

16) B)

5 23

33 115

11 41

Write an equation for the graph in the form y = a(x - h)2 + k, where a is either 1 or -1 and h and k are integers. 17) 17) 10

-10

-5

10 x

-5 -10

A) y = -(x - 3 ) 2 - 2

B) y = -(x + 3 ) 2 + 2

C) y = (x + 3 ) 2 + 2

D) y = (x + 3 ) 2 -

1 3

Provide an appropriate response. 18) What is the minimum number of x intercepts that a polynomial of degree 11 can have? Explain. A) 0 because a polynomial of odd degree may not cross the x axis at all. B) 11 because this is the degree of the polynomial. C) 1 because a polynomial of odd degree crosses the x axis at least once. D) Not enough information is given. Solve the problem. 19) To estimate the ideal minimum weight of a woman in pounds multiply her height in inches by 4 and subtract 130. Let W = the ideal minimum weight and h = height. W is a linear function of h. Find the ideal minimum weight of a woman whose height is 62 inches. A) 118 lb B) 378 lb C) 130 lb D) 120 lb

18)

19)

Find the vertex form for the quadratic function. Then find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 20) f(x) = x2 + 4x + 3 A) Standard form: f(x) = (x + 2) 2 - 1

20)

B) Standard form: f(x) = (x - 2) 2 - 1

(A) x-intercepts: - 3 , -1; y-intercept: 3 (B) Vertex (2, -1) (C) Minimum: -1 (D) y ≥ -1

(A) x-intercepts: 1, 3; y-intercept: 3 (B) Vertex (-2, -1) (C) Minimum: -1 (D) y ≥ -1

C) Standard form: f(x) = (x + 2) 2 - 1 (A) x-intercepts: - 3 , -1; y-intercept: 3 (B) Vertex (-2, -1) (C) Minimum: -1 (D) y ≥ -1

D) Standard form: f(x) = (x - 2) 2 - 1 (A) x-intercepts: - 3 , -1; y-intercept: 3 (B) Vertex (-2, -1) (C) Maximum: -1 (D) y ≤ -1

Give the domain and range of the function. 21) r(x) = x - 3 - 5 A) Domain: all real numbers; Range: all real numbers B) Domain: [- 5 , ∞); Range: all real numbers C) Domain: all real numbers; Range: [0, ∞) D) Domain: all real numbers; Range: [- 5 , ∞)

21)

Provide an appropriate response. 22) What is the maximum number of x intercepts that a polynomial of degree 10 can have? A) 9 B) 11 C) 10 D) Not enough information is given. Use a calculator to evaluate the expression. Round the result to five decimal places. 23) log (-10.25) A) -1.01072 B) 2.32728 C) 1.01072 Graph the function. 24) f(x) = 5 x

23) D) Undefined

24) y

4 2

-4

22)

-2

-2 -4

B) y

-4

-2

-4

-2

-4

D) y

-4

-2

-4

-2

-4

Solve the problem. 25) In North America, coyotes are one of the few species with an expanding range. The future population of coyotes in a region of Mississippi valley can be modeled by the equation P = 59 + 12 ∙ ln(18t + 1) , where t is time in years. Use the equation to determine when the population will reach 170. (Round your answer to the nearest tenth year.) A) 581.3 years B) 578.0 years C) 583.1 years D) 586.2 years 26) The function P, given by P(d) = 1 d + 1, gives the pressure, in atmospheres (atm), at a depth d, in 33

25)

26)

feet, under the sea. Find the pressure at 200 feet. Round your answer to the nearest whole number. A) 200 atm B) 8 atm C) 201 atm D) 7 atm Determine whether the function is linear, constant, or neither 27) y = x + 3 7 A) Linear

B) Constant

27) C) Neither

Solve graphically to two decimal places using a graphing calculator. 28) 1.7x2 - 2.6 x - 3.9 > 0 A) x < -0.93 or x > 2.46 C) -2.46 < x < 0.93

28)

B) -0.93 < x < 2.46 D) x < -2.46 or x > 0.93

Write an equation for the lowest-degree polynomial function with the graph and intercepts shown in the figure. 29) 29) y 10 5

-10

-5

-5 -10

A) f(x) = x2 + 6 x + 5 C) f(x) = x2 - 6 x + 5

B) f(x) = x2 + 5 x + 6 D) f(x) = x2 + 5 x - 6

Use the REGRESSION feature on a graphing calculator. 30) Since 1984 funeral directors have been regulated by the Federal Trade Commission. The average cost of a funeral for an adult in a Midwest city has increased, as shown in the following table.

YEAR 1980 1985 1991 1995 1996 1998 2001

AVERAGE COST OF FUNERAL $ 1926 $ 2841 $ 3842 $ 4713 $ 4830 $ 5120 $ 5340

Let x represent the number of years since 1980. Use a graphing calculator to fit a quartic function to the data. Round your answer to five decimal places. A) y = 170.5971x + 1991.5213 B) y = -0.04268 x4 + 1.53645x3 - 16.76289x2 + 231.82723x + 1927.58518 C) y = -0.04268 x4 D) y = -2.047489 x2 + 212.82699x + 1879.85469

30)

Give the domain and range of the function. 31) h(x) = -2 x A) Domain: [0, ∞); Range: [0, ∞) B) Domain: all real numbers; Range: (-∞, 0] C) Domain: (-∞, 0]; Range: all real numbers D) Domain: all real numbers; Range: (-∞, 4]

31)

Find the x-intercept(s) if they exist. 32) 6x2 = 42x A) 7

32) B) 0, 7

C) 0

D) 21

Determine whether the function is linear, constant, or neither 33) y = 2 ! 3 A) Linear

B) Constant

33) C) Neither

Graph the function using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. 34) f(x) = 4 - ln x 34) y 5

-5

A) Decreasing: (0, ∞)

B) Increasing (4, ∞) y

-5

C) Decreasing: (0, ∞)

D) Increasing (0, ∞) y

-5

Solve the equation. 35) Solve for x: 3 (1 + 2x) = 27 A) -1

35) B) 3

C) 9

D) 1

Solve the problem. 36) The U. S. Census Bureau compiles data on population. The population (in thousands) of a southern city can be approximated by P(x) = 0.08x2 - 13.08x + 927, where x corresponds to the years after 1950. In what calendar year was the population about 804,200? A) 2000 B) 1955 C) 1965

D) 1960

36)

37) The polynomial 0.0053x3 + 0.003x2 + 0.108x + 1.54 gives the approximate total earnings of a company, in millions of dollars, where x represents the number of years since 1996. This model is valid for the years from 1996 to 2000. Determine the earnings for 2000. Round to 2 decimal places. A) $2.03 million B) $2.36 million C) $2.82 million D) $2.26 million For the given function, find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 38) m(x) = -(x + 3 ) 2 + 4

37)

38)

A) (A) x-intercepts: - 5 , -1; y-intercept: -5 (B) Vertex (-3, 4) (C) Maximum: 4 (D) y ≤ 4 B) (A) x-intercepts: - 5 , -1; y-intercept: -5 (B) Vertex (-3, 4) (C) Minimum: 4 (D) y ≥ 4 C) (A) x-intercepts: - 5 , -1; y-intercept: -5 (B) Vertex (3, -4) (C) Maximum: 4 (D) y ≤ 4 D) (A) x-intercepts: 1, 5; y-intercept: -5 (B) Vertex (-3, 4) (C) Maximum: 4 (D) y ≤ 4

Solve the equation. 39) Solve for x: 24x = 8 x + 5 A) -15

39) B) 5

C) 15

D) -5

Solve the problem. 40) A country has a population growth rate of 2.4% compounded continuously. At this rate, how long will it take for the population of the country to double? Round your answer to the nearest tenth. A) 30 years B) 28.9 years C) 2.9 years D) .29 years

40)

Use the REGRESSION feature on a graphing calculator. 41) A strain of E-coli Beu-recA441 is placed into a petri dish at 30 ∘Celsius and allowed to grow. The following data are collected. Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth. The population is measured using an optical device in which the amount of light that passes through the petri dish is measured.

41)

Time in hours , x Population, y 0 0.09 2.5 0.18 3.5 0.26 4.5 0.35 6 0.50 Find the exponential equation in the form y = a ∙ b x, where x is the hours of growth. Round to four decimal places. A) y = 1.3384 x B) y = 0.0903 x C) y = 1.3384 ∙ 0.0903 x Write in terms of simpler forms. 42) logb M9 A) 9 + logb M

D) y = 0.0903 ∙ 1.3384 x

42) B) M logb 9

C) M + logb 9

D) 9 logb M

Use point-by-point plotting to sketch the graph of the equation. 43) f(x) = 2x x- 4

43)

10 5

-10

-5

10 x

5 -5 -10

B) 10

-10

-5

10 x

-10

-5

-10

10 x

D) 10

-10

-5

10 x

-10

-5

-10

10 x

The graph that follows is the graph of a polynomial function. (i) What is the minimum degree of a polynomial function that could have the graph? (ii) Is the leading coefficient of the polynomial negative or positive? 44) 44) y

6 x

-6 -10 -20

A) (i) 3 (ii) Positive

B) (i) 4 (ii) Negative

C) (i) 3 (ii) Negative

Provide an appropriate response. 45) In a profit-loss analysis, point where revenue equals cost. A) profit-loss point B) turning point C) break-even point D) inflection point Determine the domain of the function. 46) f(x) = - 7x + 9

D) (i) 4 (ii) Positive

45)

46)

A) All real numbers except 9 7

B) x ≤ 9 7

C) No solution

D) All real numbers

For the rational function below (i) Find the intercepts for the graph; (ii) Determine the domain; (iii) Find any vertical or horizontal asymptotes for the graph; (iv) Sketch any asymptotes as dashed lines. Then sketch the graph of y = f(x).

47) f(x) = -2x - 3 x+2

47) 8

-8

-4

8 x

4 -4 -8

A) (i) x intercept: - 3 ; y intercept: - 3 2 2 (ii) Domain: all real numbers except -2 (iii) Vertical asymptote: x = -2; horizontal asymptote: y = -2 (iv) 8

-8

-4

8 x

-4 -8

B) (i) x intercept: 3 ; y intercept: - 3 2 2 (ii) Domain: all real numbers except 2 (iii) Vertical asymptote: x = 2; horizontal asymptote: y = -2 (iv) 8

-8

-4

8 x

-4 -8

C) (i) x intercept: - 3 ; y intercept: - 3 2 2 (ii) Domain: all real numbers except -2 (iii) Vertical asymptote: x = -2; horizontal asymptote: y = -2 (iv) 8

-8

-4

8 x

-4 -8

D) (i) x intercept: 3 ; y intercept: - 3 2 2 (ii) Domain: all real numbers except 2 (iii) Vertical asymptote: x = 2; horizontal asymptote: y = -2 (iv) 8

-8

-4

8 x

-4 -8

Determine if the equation specifies a function with independent variable x. If so, find the domain. If not, find a value of x to which there corresponds more than one value of y. 48) x2 + y 2 = 36 48) A) A function with domain ℛ B) Not a function; for example, when x = 0, y = ±6 Determine the domain of the function. 49) f(x) = 3 - x A) x < 3 C) No solution

49) B) All real numbers except 3 D) x ≤ 3

Graph the function using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. 50) f(x) = 2 - ln(x + 4) 50) y

10 5

-10

-5

10 x

5 -5 -10

A) Decreasing: (-4, ∞) 10

B) Decreasing: (-4, ∞)

-10

-5

10 x

-10

-5

-10

C) Decreasing: (0, ∞) 10

-5

10 x

D) Decreasing: (4, ∞) y

-10

10 x

-10

-5

-10

For the polynomial function find the following: (i) Degree of the polynomial; (ii) All x intercepts; (iii) The y intercept. 51) y = 35 - x2 + 2x 51) A) (i) 2 (ii) 7, 5 (iii) 35

B) (i) 2 (ii) 7, -5 (iii) 35

C) (i) 2 (ii) 5, -7 (iii) -35

Use a calculator to evaluate the expression. Round the result to five decimal places. 52) log 51.237 A) 51.237 B) 1.70958 C) 3.93646

D) (i) 2 (ii) -5, -7 (iii) -35

52) D) Undefined

-8

-4

8 x

4 -4 -8

A) (i) x intercept: 0; y intercept: 0 (ii) Domain: all real numbers except 2 (iii) Vertical asymptote: x = 2; horizontal asymptote: y = 3 (iv) 8

-8

-4

8 x

-4 -8

B) (i) x intercept: 0; y intercept: 0 (ii) Domain: all real numbers except -2 (iii) Vertical asymptote: x = -2; horizontal asymptote: y = 3 (iv) 8

-8

-4

8 x

-4 -8

C) (i) x intercept: 0; y intercept: 0 (ii) Domain: all real numbers except -2 (iii) Vertical asymptote: x = -2; horizontal asymptote: y = -3 (iv) 8

-8

-4

8 x

-4 -8

D) (i) x intercept: 0; y intercept: 0 (ii) Domain: all real numbers except 2 (iii) Vertical asymptote: x = 2; horizontal asymptote: y = -3 (iv) 8

-8

-4

8 x

-4 -8

Find the equations of any vertical asymptotes. 54) f(x) = 4x - 11 x2 + 3x - 18 A) x = 3, x = -6

54)

B) y = 4

C) y = 3, y = -6

Determine the domain of the function. 55) f(x) = 8 x3

D) x = -3, x = 6

55)

A) All real numbers except 0 C) All real numbers

B) No solution D) x < 0

Determine whether the graph is the graph of a function. 56)

56)

10 5

-10

-5

10 x

5 -5 -10

A) function

B) not a function

Graph the function. 57) f(x) = 0.9 x

57) y

4 2

-4

-2

-2 -4

B) y

-4

-2

-4

-2

-4

D) y

-4

-2

-4

-2

-4

Solve the problem. 58) If the average cost per unit C(x) to produce x units of plywood is given by C(x) = 1200 , what is x + 40 the unit cost for 10 units? A) $80.00

B) $120.00

C) $24.00

D) $3.00

59) A professional basketball player has a vertical leap of 37 inches. A formula relating an athlete's vertical leap V, in inches, to hang time T, in seconds, is V= 48T2. What is his hang time? Round to the nearest tenth. A) 0.8 sec

B) 0.6 sec

C) 1 sec

60)

D) A = 2200 (1.03) 2t

61) If $4,000 is invested at 7% compounded annually, how long will it take for it to grow to $6,000, assuming no withdrawals are made? Compute answer to the next higher year if not exact. [A = P(1 + r)t] A) 6 years

59)

D) 0.9 sec

60) Suppose that $2200 is invested at 3% interest, compounded semiannually. Find the function for the amount of money after t years. A) A = 2200 (1.015) t B) A = 2200 (1.015) 2t C) A = 2200 (1.0125) 2t

58)

B) 5 years

C) 2 years

D) 8 years

61)

-6

A) (i) 4 (ii) Negative

B) (i) 3 (ii) Negative

C) (i) 3 (ii) Positive

D) (i) 4 (ii) Positive

Solve the equation graphically to four decimal places. 63) Let f(x) = -0.6 x2 + 3x + 1, find f(x) = 5.

63)

A) 2.5000 C) No solution

B) 4.7500 D) 2.5000 , 4.7500

Determine whether the relation represents a function. If it is a function, state the domain and range. 64) Bob Ann Dave

64)

carrots peas squash

A) function domain: {carrots, peas, squash} range: {Bob, Ann, Dave} B) function domain: {Bob, Ann, Dave} range: {carrots, peas, squash} C) not a function Find the function value. 65) Find f(4) when f(x) = 9 - 8x2. A) -119 B) -23

65) C) 137

D) -55

Solve the equation. 66) Solve for t: e -0.07t = 0.05 A) -66.4815

Round your answer to four decimal places. B) 42.7962 C) 44.321

66) D) -70.1312

Find the equation of any horizontal asymptote. 2 67) f(x) = 7x - 2x - 4 4x2 - 5 x + 6 A) y =

2 5

B) y =

67)

7 4

C) y = 0

D) None

Determine if the equation specifies a function with independent variable x. If so, find the domain. If not, find a value of x to which there corresponds more than one value of y. 68) x - y 2 = 9 68) A) A function with domain ℛ B) Not a function; for example, when x = 10, y = ±1 Solve the problem. 69) A carbon-14 dating test is performed on a fossil bone, and analysis finds that 15.5% of the original amount of carbon-14 is still present in the bone. Estimate the age of the fossil bone. (Recall that carbon-14 decays according to the equation A = A0 e -0.000124t). A) 15,035 years B) 1,500 years C) 15, 000 years D) 150 years 70) In economics, functions that involve revenue, cost and profit are used. Suppose R(x) and C(x) denote the total revenue and the total cost, respectively, of producing a new high-tech widget. The difference P(x) = R(x) - C(x) represents the total profit for producing x widgets. Given R(x) = 60x - 0.4 x2 and C(x) = 3x + 13, find P(100). A) 313

B) 2000

C) 1687

71)

72) y 4 2

-4

70)

D) 55687

71) U. S. Census Bureau data shows that the number of families in the United States (in millions) in year x is given by h(x) = 51.42 + 15.473 ∙ log x , where x = 0 is 1980. How many families were there in 2002? A) 21 million B) 48 million C) 90 million D) 72 million Graph the function. 1 x 72) f(x) = 5

69)

-2

-2 -4

B) y

-4

-2

-4

-2

-4

D) y

-4

-2

-4

-2

-4

For the polynomial function find the following: (i) Degree of the polynomial; (ii) All x intercepts; (iii) The y intercept. 73) y = x2 - 100 73) A) (i) 2 (ii) -10, 10 (iii) -100

B) (i) 1 (ii) 50 (iii) -100

C) (i) 1 (ii) 10 (iii) -100

Graph by converting to exponential form first. 74) y = log (x - 1) 2

74)

y 4 2 -4

-2

D) (i) 2 (ii) -11, 11 (iii) -100

-2 -4

B) y

-4

-2

-4

-2

-4

D) y

-4

-2

-4

-2

-4

Solve the problem. 75) If $1250 is invested at a rate of 8 1 % compounded monthly, what is the balance after 10 years? 4 [A = P(1 + i)n] A) $2281.25

B) $1594.31

C) $1031.25

D) $2844.31

Find the equations of any vertical asymptotes. 2 76) f(x) = x - 100 (x - 3 )(x + 7) A) x = 10, x = -10

75)

76)

B) y = 3, y = -7

C) x = -3

D) x = 3, x = -7

For the given function, find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 77) n(x) = -(x - 3 ) 2 + 4

77)

A) (A) x-intercepts: -5, - 1; y-intercept: -5 (B) Vertex (3, 4) (C) Maximum: 4 (D) y ≤ 4 B) (A) x-intercepts: 1, 5; y-intercept: -5 (B) Vertex (3, 4) (C) Minimum: 4 (D) y ≥ 4 C) (A) x-intercepts: 1, 5; y-intercept: -5 (B) Vertex (-3, -4) (C) Maximum: 4 (D) y ≤ 4 D) (A) x-intercepts: 1, 5; y-intercept: -5 (B) Vertex (3, 4) (C) Maximum: 4 (D) y ≤ 4

Write in terms of simpler forms. a log4 b 78) 4 A) ab

78) B) b a

C) a4b

D) b 4a

Find the equations of any vertical asymptotes. 79) f(x) = x - 1 x2 + 2 A) x = 2

79)

B) x = -2

C) x = 1, x = -1

D) None

Solve the problem. 80) A sample of 800 grams of radioactive substance decays according to the function A(t) = 800e -0.028t, where t is the time in years. How much of the substance will be left in the sample after 10 years? Round to the nearest whole gram. A) 800 grams B) 9 grams C) 605 grams

D) 1 gram

80)

For the given function, find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 81) f(x) = (x + 4) 2 - 9

81)

A) (A) x-intercepts: - 7, -1; y-intercept: 7 (B) Vertex (4, -9) (C) Minimum: -9 (D) y ≥ -9

B) (A) x-intercepts: - 7, -1; y-intercept: 7 (B) Vertex (-4, -9) (C) Minimum: -9 (D) y ≥ -9

C) (A) x-intercepts: 1, 7; y-intercept: 7 (B) Vertex (-4, -9) (C) Minimum: -9 (D) y ≥ -9

D) (A) x-intercepts: - 7, -1; y-intercept: 7 (B) Vertex (-4, -9) (C) Maximum: -9 (D) y ≤ -9

Find the vertex form for the quadratic function. Then find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 82) n(x) = -x2 + 8 x - 7 A) Standard form: n(x) = -(x - 4) 2 + 9

82)

B) Standard form: n(x) = -(x - 4) 2 + 9

(A) x-intercepts: 1, 7; y-intercept: -7 (B) Vertex (-4, -9) (C) Maximum: 9 (D) y ≤ 9

(A) x-intercepts: 1, 7; y-intercept: -7 (B) Vertex (4, 9) (C) Maximum: 9 (D) y ≤ 9

C) Standard form: n(x) = -(x + 4) 2 + 9 (A) x-intercepts: -7, - 1; y-intercept: -7 (B) Vertex (4, 9) (C) Maximum: 9 (D) y ≤ 9

D) Standard form: n(x) = -(x + 4) 2 + 9 (A) x-intercepts: 1, 7; y-intercept: -7 (B) Vertex (4, 9) (C) Minimum: 9 (D) y ≥ 9

Solve the problem. 83) To estimate the ideal minimum weight of a woman in pounds multiply her height in inches by 4 and subtract 130. Let W = the ideal minimum weight and h = height. Express W as a linear function of h. A) W(h) = 130 B) W(h) = 4h - 130 C) W(h) = 130h + 4 D) W(h) = 4 (h + 130)

83)

For the given function, find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 84) g(x) = (x - 1) 2 - 9

84)

A) (A) x-intercepts: - 2, 4; y-intercept: -8 (B) Vertex (1, -9) (C) Maximum: -9 (D) y ≤ -9

B) (A) x-intercepts: - 2, 4; y-intercept: -8 (B) Vertex (1, -9) (C) Minimum: -9 (D) y ≥ -9

C) (A) x-intercepts: - 2, 4; y-intercept: -8 (B) Vertex (-1, -9) (C) Minimum: -9 (D) y ≥ -9

D) (A) x-intercepts: -4, 2; y-intercept: -8 (B) Vertex (1, -9) (C) Minimum: -9 (D) y ≥ -9

Provide an appropriate response. 85) What is the minimum number of x intercepts that a polynomial of degree 8 can have? Explain. A) 1 because a polynomial of even degree crosses the x axis at least once. B) 8 because this is the degree of the polynomial. C) 0 because a polynomial of even degree may not cross the x axis at all. D) Not enough information is given. The graph of a function f is given. Use the graph to answer the question. 86) Use the graph of f given below to find f(10).

85)

86)

-25

-25 A) 15

B) -20

C) 0

Use a calculator to evaluate the expression. Round the result to five decimal places. 87) log 0.17 A) -1.76955 B) -4.07454 C) -0.76955

D) 10

87) D) -1.77196

Convert to a logarithmic equation. 88) 23 = 8 A) log 2 = 3 8

88) B) log 8 = 3 2

C) log2 3 = 8

D) log 8 = 2 3

Provide an appropriate response. 89) How can the graph of f(x) = - x + 1 be obtained from the graph of y = x? A) Shift it horizontally 1 units to the left. Reflect it across the x-axis. B) Shift it horizontally 1 units to the right. Reflect it across the x-axis. C) Shift it horizontally -1 units to the left. Reflect it across the x-axis. D) Shift it horizontally 1 units to the left. Reflect it across the y-axis. Use a calculator to evaluate the expression. Round the result to five decimal places. 90) log 0.234 A) 0.234 B) 1.26364 C) -1.45243

89)

90) D) -0.63074

Solve the problem. 91) In economics, functions that involve revenue, cost and profit are used. Suppose R(x) and C(x) denote the total revenue and the total cost, respectively, of producing a new high-tech widget. The difference P(x) = R(x) - C(x) represents the total profit for producing x widgets. Given R(x) = 60x - 0.4 x2 and C(x) = 3x + 13, find the equation for P(x). A) P(x) = 60x - 0.4 x2 C) P(x) = -0.4 x2 + 57x - 13

91)

B) P(x) = -0.4 x2 + 63x + 13 D) P(x) = 3x + 13

Find the function value. 92) Given that f(x) = 5 x2 - 2x, find f(t + 2). A) 5t2 - 18t + 16 B) 5t2 + 18t + 16

92) C) 3t + 6

D) t2 + 2t - 6

Determine whether the graph is the graph of a function. 93) 10

93)

-10

-5

10 x

-5 -10

A) function

B) not a function

Use the properties of logarithms to solve. 94) log (x + 3) + log x = log 54 b b b A) 6 B) -6, -3

94) C) -6

D) 3

Solve for x to two decimal places (using a calculator). 95) 700 = 500 (1.04) x A) 8.58

95)

B) 520

C) 1.35

D) 1.40

For the rational function below (i) Find any intercepts for the graph; (ii) Find any vertical and horizontal asymptotes for the graph; (iii) Sketch any asymptotes as dashed lines. Then sketch a graph of f. 96) y = 18 96) x2 - 9 y 8 6 4 2 -8 -6 -4 -2 -2

-4 -6 -8

A) (i) y intercept: 6 (ii) horizontal asymptote: y = 0; vertical asymptotes: x = 6 and x = -6 (iii) y 8 6 4 2 -8 -6 -4 -2 -2

-4 -6 -8

B) (i) y intercept: - 2 (ii) horizontal asymptote: y = 0; vertical asymptotes: x = 3 and x = -3 (iii) y 8 6 4 2 -8 -6 -4 -2 -2

-4 -6 -8

C) (i) y intercept: - 2 (ii) horizontal asymptote: y = 0 (iii) y 8 6 4 2 -8 -6 -4 -2 -2

-4 -6 -8

D) (i) y intercept: -6 (ii) horizontal asymptote: y = 0; vertical asymptotes: x = 6 and x = -6 (iii) y 8 6 4 2 -8 -6 -4 -2 -2

-4 -6 -8

Solve the problem. 97) The average weight of a particular species of frog is given by w(x) = 98 x3 , 0.1 ≤ x ≤ 0.3, where x is length (with legs stretched out) in meters and w(x) is weight in grams. (i) Describe how the graph of function w can be obtained from one of the six basic functions: y = x, y = x2, y = x3 , y = x, y =

x, or y = x . (ii) Sketch a graph of function w using part (i) as an aid.

97)

A) (i) The graph of the basic function y = x3 is vertically expanded by a factor of 98. (ii) y 3

0.1

0.2

0.3

B) (i) The graph of the basic function y = vertically expanded by a factor of 98. (ii)

x is

y 75

0.1

0.2

0.3

C) (i) The graph of the basic function y = x2 is vertically expanded by a factor of 98. (ii) y 8 6 4 2

0.1

0.2

0.3

D) (i) The graph of the basic function y = x3 is reflected on the x-axis and is vertically expanded by a factor of 98. (ii) 2

0.1

0.2

0.3

-2

-4

-8

-4

8 x

4 -4 -8

A) (i) x intercept: -5; y intercept: 3 4 (ii) Domain: all real numbers except -4 (iii) Vertical asymptote: x = -4; horizontal asymptote: y = 1 (iv) 8

-8

-4

8 x

-4 -8

B) (i) x intercept: -3; y intercept: 3 4 (ii) Domain: all real numbers except -4 (iii) Vertical asymptote: x = -4; horizontal asymptote: y = 1 (iv) 8

-8

-4

8 x

4 -4 -8

C) (i) x intercept: 5; y intercept: 3 4 (ii) Domain: all real numbers except 4 (iii) Vertical asymptote: x = 4; horizontal asymptote: y = 1 (iv) 8

-8

-4

8 x

4 -4 -8

D) (i) x intercept: 3; y intercept: 3 4 (ii) Domain: all real numbers except 4 (iii) Vertical asymptote: x = 4; horizontal asymptote: y = 1 (iv) 8

-8

-4

8 x

-4 -8

Use point-by-point plotting to sketch the graph of the equation. 99) y = x + 3 y

-5

-10

A) y

(-2, 5) 5 (0, 3)

-5

(4, -1) 5

-5

-10

B) y

-5

(-2, -1) (0, -3) -5

(4, -7) -10

99)

C) y

(4, 7) 5 (0, 3) -5

(-2, 1) 5

(4, 1) 5

-5

-10

D) y

-5 (0, -3) -5 (-2, -5)

-10

Solve the problem. 100) The following table shows a recent state income tax schedule for married couples filing a joint return in State X. State X Income Tax SCHEDULE I - MARRIED FILING JOINTLY If taxable income is Over But not over Tax due is $0 $40,000 4.25% of taxable incomes $40,000 $70,000 $3700 plus 6.75% of excess over $40,000 $70,000 $3875 plus 7.05% of excess over $70,000 (i) Write a piecewise definition for the tax due T(x) on an income of x dollars. (ii) Graph T(x). (iii) Find the tax due on a taxable income of $50,000. Of $95,000.

100)

y 8000 6000 4000 2000

20000 40000 60000 x

A) (i) T(x) =

0.0425x if 0 ≤ x ≤ 40,000 0.0675x - 1000 if 40,000 < x ≤ 70,000 0.0705x - 1060 if x > 70,000

(ii) y 8000 6000 4000 2000

20000 40000 60000 x

(iii) $2375; $5637.50

B) (i) T(x) =

0.0425x if 0 ≤ x ≤ 40,000 0.0675x - 1300 if 40,000 < x ≤ 70,000 0.0705x - 1427 if x > 70,000

(ii) y 8000 6000 4000 2000

20000 40000 60000 x

(iii) $2075; $5270.50 C) (i) T(x) =

0.0425x if 0 ≤ x ≤ 40,000 0.0675x - 1025 if 40,000 < x ≤ 70,000 0.0705x - 1375 if x > 70,000

(ii) y 8000 6000 4000 2000

20000 40000 60000 x

(iii) $2350; $5322.50

D) (i) 0.0425x if 0 ≤ x ≤ 40,000 0.0675x - 990 if 40,000 < x ≤ 70,000 0.0705x - 1000 if x > 70,000

T(x) = (ii)

y 8000 6000 4000 2000

20000 40000 60000 x

(iii) $2385; $5697.50

-10

A) (i) 3 (ii) Negative

B) (i) 2 (ii) Negative

C) (i) 2 (ii) Positive

D) (i) 3 (ii) Positive

Determine if the equation specifies a function with independent variable x. If so, find the domain. If not, find a value of x to which there corresponds more than one value of y. 102) y = x 2 - 8 102) A) A function with domain ℛ B) Not a function; for example, when x = -8, then y = ±1 Use point-by-point plotting to sketch the graph of the equation.

103) y = x2 + 4

103) y

10 5

-10

-5

10 x

5 -5 -10

B) 10

-10

-5

10 x

-10

-5

-10

10 x

D) 10

-10

-5

10 x

-10

-5

-10

Determine if the equation specifies a function with independent variable x. If so, find the domain. If not, find a value of x to which there corresponds more than one value of y. 104) x2 - y 2 = 9 104) A) A function with domain all real numbers except x = 5 B) Not a function; for example, when x = 5, y = ±4 Graph by converting to exponential form first.

105) y = log

(x + 4)

105) y

4 2 -4

-2

-2 -4

B) y

-4

-2

-4

-2

-4

D) y

-4

-2

-4

-2

-4

Solve for x to two decimal places (using a calculator). 106) 5.2 = 1.006 12x A) 1.07

106)

B) 2.32

C) 5.17

Write in terms of simpler forms. 107) log XY 4 A) log X + log Y 4 4 C) log X + log Y 2 2

D) 22.97

107) B) log X - log Y 2 2 D) log X - log Y 4 4

Use the properties of logarithms to solve. 108) log x + log (x - 2) = log 24 7 7 7 A) 7 B) 2

108) C) 24

D) 6

Graph the function. 109) f(x) = 2- x - 3

109) 5 4 3 2 1

-5 -4 -3 -2 -1 -1

1 2 3 4 5 x

-2 -3 -4 -5

B) 5 4 3 2 1

5 4 3 2 1 1 2 3 4 5 x

-5 -4 -3 -2 -1 -1

1 2 3 4 5 x

-5 -4 -3 -2 -1 -1

-2 -3 -4 -5

D) 5 4 3 2 1 -5 -4 -3 -2 -1 -1

5 4 3 2 1 1 2 3 4 5 x

-5 -4 -3 -2 -1 -1

-2 -3 -4 -5

1 2 3 4 5 x

-2 -3 -4 -5

Determine whether the function is linear, constant, or neither 110) y - 12 = 0 A) Linear B) Constant Solve the problem. 40

110) C) Neither

111) A retail chain sells washing machines. The retail price p(x) (in dollars) and the weekly demand x for a particular model are related by the function p(x) = 625 - 5 x, where 50 ≤ x ≤ 500. (i) Describe how the graph of the function p can be obtained from the graph of one of the six basic 3 functions: y = x, y = x2, y = x3 , y = x, y = x, or y = x . (ii) Sketch a graph of function p using part (i) as an aid. y

A) (i) The graph of the basic function y = vertically expanded by a factor of 625, and shifted up 5 units. (ii)

x is

16000 12800 9600 6400 3200

100200300400500

B) (i) The graph of the basic function y = vertically expanded by a factor of 5, and shifted up 625 units. (ii) 1000

x is

800 600 400 200 100 200 300 400

111)

C) (i) The graph of the basic function y = reflected in the x-axis and vertically expanded by a factor of 5. (ii) 100

x is

50 100 200 300 400

-50 -100 -150

D) (i) The graph of the basic function y = x is reflected in the x-axis, vertically expanded by a factor of 5, and shifted up 625 units. (ii) 1000

800 600 400 200 100 200 300 400

112) An initial investment of $12,000 is invested for 2 years in an account that earns 4% interest, compounded quarterly. Find the amount of money in the account at the end of the period. A) $12,979.20 B) $12,865.62 C) $12,994.28 D) $994.28 Graph the function.

112)

113)

113) f(x) =

x- 1 -4

if x < 1 if x ≥ 1 y 5

-5

B) y

-5

D) y

-5

Use the properties of logarithms to solve. 114) ln (3x - 4) = ln 20 - ln (x - 5) A) 5, 5 B) 0, 19 3 3

114) 19 C) 3

D) -5, - 19 3

Provide an appropriate response. 115) How can the graph of f(x) = -(x -1 ) 2 6 be obtained from the graph of y = x2? A) Shift it horizontally 1 units to the right. Reflect it across the y-axis. Shift it 6 units up. B) Shift it horizontally 1 units to the right. Reflect it across the x-axis. Shift it 6 units up. C) Shift it horizontally 1 units to the left. Reflect it across the x-axis. Shift it 6 units up. D) Shift it horizontally 1 units to the right. Reflect it across the y-axis. Shift it 6 units down. Find the vertex form for the quadratic function. Then find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 116) g(x) = x2 - 2x - 3

116)

A) Standard form: g(x) = (x - 1) 2 - 4 (A) x-intercepts: - 1, 3; y-intercept: -3 (B) Vertex (-1, -4) (C) Minimum: -4 (D) y ≥ -4

B) Standard form: g(x) = (x + 1) 2 - 4 (A) x-intercepts: -3, 1; y-intercept: -3 (B) Vertex (1, -4) (C) Minimum: -4 (D) y ≥ -4

C) Standard form: g(x) = (x + 1) 2 - 4 (A) x-intercepts: - 1, 3; y-intercept: -3 (B) Vertex (1, -4) (C) Maximum: -4 (D) y ≤ -4

D) Standard form: g(x) = (x - 1) 2 - 4 (A) x-intercepts: - 1, 3; y-intercept: -3 (B) Vertex (1, -4) (C) Minimum: -4 (D) y ≥ -4

Solve the problem. 117) Hi-Tech UnWater begins a cable TV advertising campaign in Miami to market a new water. The percentage of the target market that buys water is estimated by the function w(t) = 100(1 - e -0.02t), t represents the number of days of the campaign. After how long will 90% of the target market have bought the water? A) 120 days B) 90 days

C) 115 days

117)

D) 3 days

Convert to a logarithmic equation. 118) 5 2 = 25 A) 25 = log5 2

115)

118) B) 5 = log2 25

C) 2 = log 5 25

D) 2 = log25 5

Graph the function using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. 119) f(x) = 3 ln x 119) y 5

-5

A) Increasing: (0, ∞)

B) Increasing: (-3, ∞) y

-5

C) Decreasing: (0, ∞)

D) Decreasing: (0, ∞) y

-5

Solve the problem. 120) The level of a sound in decibels (db) is determined by the formula N = 10 ∙ log(I × 10 12) db, where I is the intensity of the sound in watts per square meter. A certain noise has an intensity of 8.49 × 10 -4 watts per square meter. What is the sound level of this noise? (Round your answer to the nearest decibel.) A) 206 db

B) 9 db

C) 89 db

D) 79 db

120)

121) Assume that a person's critical weight W, defined as the weight above which the risk of death h 3 rises dramatically, is given by W(h) = , where W is in pounds and h is the person's height 11.9

121)

in inches. Find the tcritical weight for a person who is 6 ft 11 in. tall. Round to the nearest tenth. A) 377.4 lb B) 221.5 lb C) 339.3 lb D) 212.4 lb Graph the function. 122) f(x) = 4(x - 3 ) - 2

122) y

4 2 -4

-2

-2 -4

B) y

-4

-2

-4

-2

-4

D) y

-4

-2

-4

-2

-4

Write an equation for a function that has a graph with the given transformations. 123) The shape of y = x is shifted 5 units to the left. Then the graph is shifted 7 units upward. A) f(x) = x + 7 + 5 B) f(x) = x - 5 + 7 C) f(x) = 7 x + 5 D) f(x) = x + 5 + 7

123)

Evaluate. 124) log 8 4 8

124)

A) 8 4

B) 4

C) 8

D) 32

Determine whether the relation represents a function. If it is a function, state the domain and range. 125) {(41, -2), (5, -1), (5, 0), (6, 1), (14, 3)} A) function domain: {-2, -1, 0, 1, 3} range: {41, 6, 5, 14} B) function domain: {41, 6, 5, 14} range: {-2, -1, 0, 1, 3} C) not a function Solve the problem. 126) Financial analysts in a company that manufactures ovens arrived at the following daily cost equation for manufacturing x ovens per day: C(x) = x2 + 4x + 1800. The average cost per unit at a production level of x ovens per day is C(x) = C(x)/x. (i) Find the rational function C. (ii) Sketch a graph of C(x) for 10 ≤ x ≤ 125. (iii) For what daily production level (to the nearest integer) is the average cost per unit at a minimum, and what is the minimum average cost per oven (to the nearest cent)? HINT: Refer to the sketch in part (ii) and evaluate C(x) at appropriate integer values until a minimum value is found. y

125)

126)

2 A) (i) C(x) = x + 4x + 1800 x

2 B) (i) C(x) = x + 4x + 1800 x

(ii)

200

400

160

320

120

240

160

20 40 60 80 100 120 x

(iii) 42 units; $88.86 per oven 2 C) (i) C(x) = x + 4x + 1800 x

(iii) 61 units; $133.29 per oven 2 D) (i) C(x) = x + 4x + 1800 x

(ii)

200

500

160

400

120

300

200

100 20 40 60 80 100 120 x

20 40 60 80 100 120 x

(iii) 22 units; $48.93 per oven

(iii) 44 units; $185.61 per oven

Use the properties of logarithms to solve. 127) log (x + 10) - log (x + 4) = log x A) 2, - 5 B) -5

127) C) 2

D) 6

Solve the problem. 128) Suppose the cost per ton, y, to build an oil platform of x thousand tons is approximated by C(x) = 212,500 . What is the cost per ton for x = 30? x + 425 A) $16.67

B) $467.03

C) $425.00

D) $7083.33

Find the equations of any vertical asymptotes. 2 129) f(x) = x + 3x x2 - 6x - 27 A) x = -9, x = 3

128)

129)

B) x = 9, x = -3

C) x = 9

D) None

Convert to an exponential equation. 130) log 27 = 3 9 2 A) 27 = 9 3/2

130) B) 27 =

3 9 2

3 = 9 27 2

D) 9 = 273/2

Find the equation of any horizontal asymptote. 2 131) f(x) = 5x + 5 5x2 - 5 A) y = -5

131)

B) y = 1

C) y = 5

D) None

Determine if the equation specifies a function with independent variable x. If so, find the domain. If not, find a value of x to which there corresponds more than one value of y. 132) xy = -5 132) A) A function with domain all real numbers except x = 0 B) Not a function; for example, when x = -5, y = ±1 Solve the problem. 133) The number of books in a community college library increases according to the function B = 7200e 0.03t, where t is measured in years. How many books will the library have after 8 year(s)? A) 10,275

B) 4462

C) 9153

D) 7200

Use the properties of logarithms to solve. 134) log 6 (4x - 5) = 1 A) 7

134)

11 6

log 5 4

11 4

Graph the function. 135) Assume it costs 25 cents to mail a letter weighing one ounce or less, and then 20 cents for each additional ounce or fraction of an ounce. Let L(x) be the cost of mailing a letter weighing x ounces. Graph y = L(x). Use the interval (0, 4]. A) y 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1 2 3 Weight (in ounces)

133)

135)

B) y 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1 2 3 Weight (in ounces)

1 2 3 Weight (in ounces)

C) y 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

D) y 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1 2 3 Weight (in ounces)

Solve the problem. 136) Assume that a savings account earns interest at the rate of 2% compounded monthly. If this account contains $1000 now, how many months will it take for this amount to double if no withdrawals are made? A) 450 months B) 417 months C) 408 months D) 12 months

136)

Solve the equation graphically to four decimal places. 137) Let f(x) = -0.7x2 + 2x + 3, find f(x) = 2.

137)

A) -0.4341 C) No solution

B) 3.2912 D) -0.4341, 3.2912

Solve the problem. 138) The point at which a company's costs equals its revenue is the break-even. C represents cost, in dollars, of x units of a product. R represents the revenue, in dollars, for the sale of x units. Find the number of units that must be produced and sold in order to break even. C = 15x + 12,000 R = 18x - 6000 A) 545 B) 6000 C) 12,000 D) 800

138)

Solve the equation. 139) Solve for x: (e x) x ∙ e 6 = e 5x A) {-3, -2} B) {2}

139) C) {3, 2}

D) {3}

Solve the problem. 140) Under certain conditions, the power P, in watts per hour, generated by a windmill with winds blowing v miles per hour is given by P(v) = 0.015v 3 . Find the power generated by 18 -mph winds. A) 58.32 watts per hour C) 4.86 watts per hour

140)

B) 0.00006075 watts per hour D) 87.48 watts per hour

-10

A) (i) 2 (ii) Positive

B) (i) 2 (ii) Negative

C) (i) 3 (ii) Positive

D) (i) 3 (ii) Negative

142) f(x) = x + 2 x+1

142) 4

3 2 1 -4 -3 -2 -1 -1

4 x

-2 -3 -4

A) (i) x intercept: 2; y intercept: 2 (ii) Domain: all real numbers except 1 (iii) Vertical asymptote: x = 1; horizontal asymptote: y = 1 (iv) 4

3 2 1 -4 -3 -2 -1 -1

4 x

-2 -3 -4

B) (i) x intercept: 0; y intercept: 0 (ii) Domain: all real numbers except -1 (iii) Vertical asymptote: x = -1; horizontal asymptote: y = 1 (iv) 4

3 2 1 -4 -3 -2 -1 -1

4 x

-2 -3 -4

C) (i) x intercept: 0; y intercept: 0 (ii) Domain: all real numbers except 1 (iii) Vertical asymptote: x = 1; horizontal asymptote: y = 1 (iv) 4

3 2 1 -4 -3 -2 -1 -1

4 x

-2 -3 -4

D) (i) x intercept: -2; y intercept: 2 (ii) Domain: all real numbers except -1 (iii) Vertical asymptote: x = -1; horizontal asymptote: y = 1 (iv) 4

3 2 1 -4 -3 -2 -1 -1

4 x

-2 -3 -4

Determine if the equation specifies a function with independent variable x. If so, find the domain. If not, find a value of x to which there corresponds more than one value of y. 143) xy + 3y = 1 143) A) A function with domain all real numbers except x = -3 B) Not a function; for example, when x = 1, y = ±3 Graph the function using a calculator and point-by-point plotting. Indicate increasing and decreasing intervals. 144) f(x) = 4 ln x 144) y 5

-5

A) Decreasing: (0, ∞)

B) Decreasing: 0, y

Increasing:

1 2

1 ,∞ 2 ) y 5

-5

C) Decreasing: (0, 4] Increasing: [4, ∞)

D) Decreasing: (0, 1] Increasing: [1, ∞) y

-5

For the polynomial function find the following: (i) Degree of the polynomial; (ii) All x intercepts; (iii) The y intercept. 145) f(x) = (x6 + 7)(x10 + 9) 145) A) (i) 60 (ii) none (iii) -63

B) (i) 60 (ii) 7, 9 (iii) -63

C) (i) 16 (ii) 7, 9 (iii) 63

D) (i) 16 (ii) none (iii) 63

Write an equation for the graph in the form y = a(x - h)2 + k, where a is either 1 or -1 and h and k are integers. 146) 146) 10

-10

-5

10 x

-5 -10

A) y = (x - 5 ) 2 - 6 C) y = (x + 6 ) 2 + 5

B) y = (x - 6 ) 2 - 5 D) y = (x - 6 ) 2 - 6

Use the REGRESSION feature on a graphing calculator. 147) The total cost of the Democratic and the Republican national conventions has increased 596% over the 20 -year period between 1980 and 2004. The following table lists the total cost, in millions of dollars, for selected years. Year, x Cost, y 1980, x = 0 $ 23.1 1984, x = 4 31.8 1988, x = 8 44.4 1992, x = 12 58.8 1996, x = 16 90.6 2000, x = 20 160.8 2004, x = 24 170.5 Find the exponential functions that best estimates this data. Round your answer to four decimal places A) y = 22.2887 ∙ (1.0929) x B) y = 1.0929 ∙ (22.2887) x C) y = 22.2887x∙ (1.0929) x

D) y = 6.6643x + 2.8857

147)

-6

A) (i) 1 (ii) Negative

B) (i) 1 (ii) Positive

C) (i) 2 (ii) Positive

D) (i) 2 (ii) Negative

Solve the problem. 149) The function M described by M(x) = 2.89x + 70.64 can be used to estimate the height, in centimeters, of a male whose humerus (the bone from the elbow to the shoulder) is x cm long. Estimate the height of a male whose humerus is 30.93 cm long. Round your answer to the nearest four decimal places. A) 157.3400 m B) 156.5375 cm C) 160.0277 cm D) 30.9300 cm 150) The function F described by F(x) = 2.75x + 71.48 can be used to estimate the height, in centimeters, of a woman whose humerus (the bone from the elbow to the shoulder) is x cm long. Estimate the height of a woman whose humerus is 30.93 cm long. Round your answer to the nearest four decimal places. A) 43.3000 cm B) 156.5375 cm C) 105.1600 cm D) 13.5775 cm Use the REGRESSION feature on a graphing calculator. 151) The average retail price in the Spring of 2000 for a used Camaro Z28 coupe depends on the age of the car as shown in the following table.

149)

150)

151)

Age, x 1 2 3 4 5 6 7 8 9 Price, y 18,325 15,925 13,685 11,805 10,490 8885 8015 6480 5710 Find the quadratic model that best estimates this data. Round your answer to whole numbers. A) y = 102x2 - 2576x + 20,669 B) y = -1551x + 18,790x C) y = 102x2 - 2576x

D) y = -9x3 + 235 x2 - 3134x + 21,252

Solve the problem. 152) The population P, in thousands, of Fayetteville is given by P(t) = 300t , where t is the time, in 2t2 + 7 months. Find the population at 9 months. A) 7988 B) 40,000

C) 30, 769

D) 15, 976

152)

Determine whether the relation represents a function. If it is a function, state the domain and range. 153)

153)

3 → 15 4 → 20 5 → 25 6 → 30 A) function domain:{15, 20, 25, 30} range: {3, 4, 5, 6} B) function domain: {3, 4, 5, 6} range: {15, 20, 25, 30} C) not a function Convert to a logarithmic equation. 154) 10 0.4771 = 3

154)

A) 0.4771 = log 9 10

B) 0.4771 = log 10

C) 0.4771 = log 3

D) 3 = log 0.4771

Find the range of the given function. Express your answer in interval notation. 155) f(x) = 4x2 + 16x + 19 A) [3, ∞)

B) [ - 2, ∞)

C) (-∞, -3]

Use a calculator to evaluate the expression. Round the result to five decimal places. 156) ln 0.027 A) 0.56864 B) -3.61192 C) -1.56864

155) D) (-∞, 2]

156) D) Undefined

Give the domain and range of the function. 157) g(x) = x2 - 4

157)

A) Domain: [0, ∞); Range: [0, ∞) B) Domain: all real numbers; Range: [-4, ∞) C) Domain: all real numbers; Range: [2, ∞) D) Domain: [4, ∞); Range: all real numbers 158) s(x) = 5 - x A) Domain: ( 5, ∞); Range: (-∞, 0] B) Domain: (-∞, 5]; Range: [0, ∞) C) Domain: all real numbers; Range: [0, ∞) D) Domain: (-∞, 5) ∪ (5, ∞); Range: (-∞, 0) ∪ (0, ∞)

158)

Determine whether there is a maximum or minimum value for the given function, and find that value. 159) f(x) = - x2 - 18x - 90 A) Maximum: - 9 C) Minimum: 9

B) Minimum: -9 D) Minimum: 0

Graph the function. 57

159)

160)

160) f(x) =

-x + 3 2x - 3

if x < 2 if x ≥ 2 y 5

-5

B) y

-5

D) y

-5

Use the REGRESSION feature on a graphing calculator. 161) As the number of farms has decreased in South Carolina, the average size of the remaining farms has grown larger, as shown below.

161)

AVERAGE ACREAGE PER FARM 127 119 135 137 155 196 283 353 406 440 420

YEAR 1900 (x = 0) 1910 (x = 10) 1920 1930 1940 1950 1960 1970 1980 1990 2000 (x = 100)

Let x represent the number of years since 1900. Use a graphing calculator to fit a quadratic function to the data. Round your answer to five decimal places. A) y = 0.02536x2 + 1.21114 x + 102.58741 B) y = -.00114x3 + 0.19605x2 - 5.29775 + 143.55245 C) y = 0.02536x3 + 1.21114 + 102.58741 D) y = 0.02536x3 + 1.21114 x + 102.58741 Determine the domain of the function. 162) f(x) = x x- 2

162)

A) x < 2 C) All real numbers except 2

B) No solution D) All real numbers

Give the domain and range of the function. 163) f(x) = x2 + 2

163)

A) Domain: [0, ∞); Range: [0, ∞) B) Domain: all real numbers; Range: [2, ∞) C) Domain: [2, ∞); Range: all real numbers D) Domain: all real numbers; Range: [5, ∞) Determine whether there is a maximum or minimum value for the given function, and find that value. 164) f(x) = x2 - 20x +104 A) Maximum: -4

B) Minimum: 0

C) Minimum: 4

D) Maximum: 10

164)

Determine whether the graph is the graph of a function. 165)

165)

10 5

-10

-5

10 x

5 -5 -10

A) function

B) not a function

Write an equation for the lowest-degree polynomial function with the graph and intercepts shown in the figure. 166) 166) y 10 5

-10

-5

-5 -10

A) f(x) = -x2 - 11x - 30 C) f(x) = x2 + 30 x + 11

B) f(x) = x2 + 30 x - 11 D) f(x) = x2 + 11x + 30

Convert to an exponential equation. 167) ln 44 = 3.7842 A) e 3.7842 = 44

167) B) e 3.7842 = 1

C) e 3.7842 = ln 44

D) e 44 = 3.7842

Write in terms of simpler forms. 168) log x b s A) log x - s b

168) C) log x 2b s

B) log x - log s b b

D) log x + log s b b

Solve the problem. 169) The number of reports of a certain virus has increased exponentially since 1960. The current number of cases can be approximated using the function r(t) = 207 e 0.005t, where t is the number of years since 1960. Estimate the of cases in the year 2010. A) 240 B) 190 C) 207 Use the properties of logarithms to solve. 170) log (x - 9) = 1 - log x A) -1, 10 B) -10, 1

D) 266

170) C) -10

D) 10

Find the vertex form for the quadratic function. Then find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 171) m(x) = -x2 - 8 x - 7 A) Standard form: m(x) = -(x + 4) 2 + 9

171)

B) Standard form: m(x) = -(x - 4) 2 + 9

(A) x-intercepts: - 7, -1; y-intercept: -7 (B) Vertex (-4, 9) (C) Maximum: 9 (D) y ≤ 9

(A) x-intercepts: 1, 7; y-intercept: -7 (B) Vertex (-4, 9) (C) Maximum: 9 (D) y ≤ 9

C) Standard form: m(x) = -(x - 4) 2 + 9 (A) x-intercepts: - 7, -1; y-intercept: -7 (B) Vertex (-4, 9) (C) Minimum: 9 (D) y ≥ 9

D) Standard form: m(x) = -(x + 4) 2 + 9 (A) x-intercepts: - 7, -1; y-intercept: -7 (B) Vertex (4, -9) (C) Maximum: 9 (D) y ≤ 9

Determine whether the function is linear, constant, or neither 172) y = x3 - x2 + 8 A) Linear

169)

172)

B) Constant

C) Neither

Use the properties of logarithms to solve. 173) log x - log 5 = log 2 - log (x - 3) b b b b A) 2, 5 B) 5

173) C) 3

D) 2

Write an equation for a function that has a graph with the given transformations. 174) The shape of y = x2 is vertically stretched by a factor of 10, and the resulting graph is reflected

174)

across the x-axis. A) f(x) = (x - 10) 2 C) f(x) = 10 x2

B) f(x) = 10 (x - 10) 2 D) f(x) = - 10 x2

Solve the problem. 175) Book sales on the Internet (in billions of dollars) in year x are approximated by f(x) = 1.84 + 2.1 ∙ ln x, where x = 0 corresponds to 2000. How much will be spent on Internet book sales in 2008? Round to the nearest tenth. A) 8.0 billion B) 6.2 billion C) 3.9 billion D) 6.0 billion 61

175)

Use a calculator to evaluate the expression. Round the result to five decimal places. 176) ln 1097 A) 4.69775 B) 3.04021 C) 9.30292

176) D) 7.00033

Sketch the graph of the function. 177) f(x) = x + 1 x2 + x - 2

177)

B) 8 6 4 2 -8 -6 -4 -2 -2

8 6 4 2 2 4 6 8 10

-8 -6 -4 -2 -2

-4 -6 -8

2 4 6 8 10

-4 -6 -8

D) 8 6 4 2 -8 -6 -4 -2 -2

8 6 4 2 2 4 6 8 10

-8 -6 -4 -2 -2

-4 -6 -8

2 4 6 8 10

-4 -6 -8

Compute and simplify the difference quotient f(x + h) - f(x) , h ≠ 0. h 178) f(x) = 5 x2 + 7x A) 10x + 7

178) C) 10x2 + 5h+ 7x

B) 15x - 7h + 14

D) 10x + 5h + 7

Convert to an exponential equation. 179) log 8 512 = t A) 8 512 = t

179) B) 5128 = t

C) 8 t = 512

D) t8 = 512

Solve graphically to two decimal places using a graphing calculator. 180) 1.5 x2 - 4.7x - 2.9 ≤ 0 A) x < -3.66 or x > 0.53 C) -3.66 < x < 0.53

180)

B) x < -0.53 or x > 3.66 D) -0.53 < x < 3.66

Find the x-intercept(s) if they exist. 181) x2 + 6x + 5 = 0 A) 10, -5

181) B)

5, -

C) 1 , 5

D) -1, -5

Determine whether the relation represents a function. If it is a function, state the domain and range. 182) {(-2, 2), (-1, -1), (0, -2), (1, -1), (3, 7)} A) function domain: {-2, -1, 0, 1, 3} range: {2, -1, -2, 7} B) function domain: {2, -1, -2, 7} range: {-2, -1, 0, 1, 3} C) not a function Solve the problem. 183) Since life expectancy has increased in the last century, the number of Alzheimer's patients has increased dramatically. The number of patients in the United States reached 4 million in 2000. Using data collected since 2000, it has been found that the data can be modeled by the exponential function y = 4.19549 ∙ (1.02531) x, where x is the years since 2000. Estimate the Alzheimer's patients in 2025. Round to the nearest tenth. A) 8.0 million B) 7.8 million C) 4.8 million

182)

183)

D) 3.9 million

For the polynomial function find the following: (i) Degree of the polynomial; (ii) All x intercepts; (iii) The y intercept. 184) y = x2 + 3 x - 18 184) A) (i) 2 (ii) 6, -3 (iii) -18

B) (i) 2 (ii) -6, 1 (iii) -18

C) (i) 2 (ii) 6, 3 (iii) -18

D) (i) 2 (ii) -6, 3 (iii) -18

Sketch the graph of the function. x2 185) f(x) = x2 - x - 6

185)

B) 8 6 4 2 -8 -6 -4 -2 -2

8 6 4 2 2 4 6 8 10

-8 -6 -4 -2 -2

-4 -6 -8

2 4 6 8 10

-4 -6 -8

D) 8 6 4 2 -8 -6 -4 -2 -2

8 6 4 2 2 4 6 8 10

-8 -6 -4 -2 -2

-4 -6 -8

2 4 6 8 10

-4 -6 -8

For the polynomial function find the following: (i) Degree of the polynomial; (ii) All x intercepts; (iii) The y intercept. 186) y = (x + 4)(x + 2)(x + 1) 186) A) (i) 3 B) (i) 3 C) (i) 3 D) (i) 3 (ii) -4, -2, -1 (ii) 4, 2, 1 (ii) -4, -2, -1 (ii) 4, 2, 1 (iii) -2 (iii) 8 (iii) 8 (iii) 2

187) y = 8x + 5 A) (i) 1

187) B) (i) 1 (ii) 5

(ii) - 5 8

(iii) 5 8

(iii) 5

C) (i) 1 (ii) 5 8

D) (i) 1

(iii) 5

(iii) 8

(ii) -

8 5

Solve the equation graphically to four decimal places. 188) Let f(x) = -0.4x2 + 2x + 3, find f(x) = -5.

188)

A) -2.6235 , 7.6235 C) No solution

B) -2.6235 D) 7.6235

Use a calculator to evaluate the expression. Round the result to five decimal places. 189) log 36.8 8 A) 0.57674 B) 1.73388 C) 1.56585

189) D) 3.60550

Write an equation for the lowest-degree polynomial function with the graph and intercepts shown in the figure. 190) 190) y

1 2 3 4 5 6x

-6 -5 -4 -3 -2 -1

A) f(x) = -x3 - 16x C) f(x) = -x3 - 16x

B) f(x) = x3 + 16x D) f(x) = -x3 + 16x

For the following problem, (i) graph f and g in the same coordinate system; (ii) solve f(x) = g(x) algebraically to two decimal places; (iii) solve f(x) > g(x) using parts i and ii; (iv) solve f(x) < g(x) using parts i and ii. 191) f(x) = -0.8x(x - 8), g(x) = 0.4x + 3.2; 0 ≤ x ≤ 10 191) 12 11 10 9 8 7 6 5 4 3 2 1

9 x

A) (i) f is the curve, g is the line 12 11 10 9 8 7 6 5 4 3 2 1

9 x

(ii) 0.58, 7.98 (iii) 0.58 < x < 7.98 (iv) 0 ≤ x < 0.58 or 7.98 < x ≤ 8 B) (i) f is the curve, g is the line 12 11 10 9 8 7 6 5 4 3 2 1

(ii) 0.58, 6.92 (iii) 0.58 < x < 6.92 (iv) 0 ≤ x < 0.58 or 6.92 < x ≤ 8

C) (i) f is the curve, g is the line 12 11 10 9 8 7 6 5 4 3 2 1

9 x

(ii) 0.61, 7.02 (iii) 0.61 < x < 7.02 (iv) 0 ≤ x < 0.61 or 7.02 < x ≤ 8 D) (i) f is the curve, g is the line 12 11 10 9 8 7 6 5 4 3 2 1

(ii) 0.61, 7.98 (iii) 0.61 < x < 7.98 (iv) 0 ≤ x < 0.61 or 7.98 < x ≤ 8

Find the equation of any horizontal asymptote. 2 192) f(x) = x + 3 x - 3 x- 3 A) y = 4

192)

B) None

C) y = -3

D) y = 3

Find the range of the given function. Express your answer in interval notation. 193) f(x) = -2x2 + 12x - 23 A) [5, ∞)

B) (-∞, -5]

C) (-∞, -3]

193) D) [-3, ∞)

Answer Key Testname: CHAP 02_14E

1) 53 2) Basic function is f(x) = x2; shift right 2 units, shift up 5 units. f(x) = (x - 2) 2 + 5 3) -5 4) f(x) = -(x + 2) 2 + 9 ; vertex: (-2, 9); maximum: f(-2) = 9; Range of f = {y y ≤ 9 } ; y-intercept: (0, 5); x-intercepts: (-5, 0), (1, 0). 5) g(x) =

x- 9

x< 7

(x - 5) 2 - 1

x≥7 x 6) y = 1.1389(1.0429 ); $7.54; $9.30 7) P(x) = -6x2 + 80.3x - 150, must sell approximately 6.69 million cameras. 8) Max f(x) = 25 4 8

6 4 2

5 x

-5 -2

9) 2,018 bacteria 10) (A) C(x) = 1500 + 105 + x x (B) 1500

1000

500

100

(C) 39; $182.46 11) -27, -12, - 33 2 12) Basic function is f(x) = x ; reflect over the x -axis, shift left 4 units, shift down 2 units. f(x) = - x + 4 -2

Answer Key Testname: CHAP 02_14E

13) Choice (A) defines a function. To each element (student) of the first set (or domain), there corresponds exactly one element (teacher) of the second set (or range). Choice (B) does not define a function. An element (student) of the first set (or domain) corresponds to more that one element (teacher) of the second set (or range). 14) f(x) = 2x has domain all real numbers except x = 48. 48 - x 15) D 16) C 17) B 18) C 19) A 20) C 21) D 22) C 23) D 24) D 25) B 26) D 27) A 28) A 29) C 30) B 31) B 32) B 33) B 34) A 35) D 36) D 37) B 38) A 39) C 40) B 41) D 42) D 43) B 44) D 45) C 46) D 47) C 48) B 49) D 50) B 51) B 52) B 53) A 54) A 55) A 56) B 57) B 69

Answer Key Testname: CHAP 02_14E

58) C 59) D 60) B 61) A 62) B 63) C 64) C 65) A 66) B 67) B 68) B 69) A 70) C 71) D 72) C 73) A 74) C 75) D 76) D 77) D 78) B 79) D 80) C 81) B 82) B 83) B 84) B 85) C 86) B 87) C 88) B 89) A 90) D 91) C 92) B 93) A 94) A 95) A 96) B 97) A 98) D 99) C 100) A 101) C 102) A 103) D 104) B 105) A 106) D 70

Answer Key Testname: CHAP 02_14E

107) A 108) D 109) B 110) B 111) D 112) C 113) D 114) C 115) B 116) D 117) C 118) C 119) A 120) C 121) C 122) D 123) D 124) B 125) C 126) A 127) C 128) B 129) C 130) A 131) B 132) A 133) C 134) D 135) C 136) B 137) D 138) B 139) C 140) D 141) C 142) D 143) A 144) D 145) D 146) D 147) A 148) A 149) C 150) B 151) A 152) D 153) B 154) C 155) A 71

Answer Key Testname: CHAP 02_14E

156) B 157) B 158) B 159) A 160) B 161) A 162) C 163) B 164) C 165) B 166) A 167) A 168) B 169) D 170) D 171) A 172) C 173) B 174) D 175) B 176) D 177) D 178) D 179) C 180) D 181) D 182) A 183) B 184) D 185) D 186) C 187) A 188) A 189) B 190) D 191) B 192) B 193) B

CHAPTER 3

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) An investor purchased 150 shares of a stock at $15.80 per share. The investor holds the stock for 39 weeks and then sells the stock for $19.25 per share. Use the commission schedule for this company given below to find the annual rate of interest earned by this investor. Express your answer as a percentage, correct to one decimal place. Principle Under $2500 $2500 - $7500 Over $7500

Commission $25 + 1.6% of principle $38 + 1.1% of principle $105 + 0.5% of principle

Solve the problem. 2) You can afford monthly deposits of $200 into an account that pays 8% compounded monthly. How many months will it be until you have $15,000 to buy a car? (Round up to the next higher month if not an integer.) 3) A couple decides on the following savings plan for their child's college education. When the child is 6 months old, and every 6 months thereafter, they will deposit $310 into a savings account paying 9.5% interest compounded semi-annually. After the child's tenth birthday, having made 20 such payments, they will stop making deposits and let the accumulated money earn interest, at the same rate, for 8 more years, until the child is 18 years old and ready for college. How much money (to the nearest dollar) will be in the account when the child is ready for college? Provide an appropriate response. 4) How much should you invest now at 6% compounded semiannually to have $8,500 to buy a car in 2.5 years? Solve the problem. Round to the nearest cent as needed. 5) You have decided to buy a new stereo system for $2,500 and agreed to pay in 30 equal quarterly payments at 1.25% interest per quarter on the unpaid balance. How much are your payments? Use an amortization table to solve the problem. Round to the nearest cent. 6) A $90,000 home was financed by making a 20% down payment and signing a 30 -year mortgage at 6.25% annual interest compounded monthly for the unpaid balance. The first payment is $443.32. How much of the first month's payment will apply towards reducing the principal? Provide an appropriate response. 7) What amount will be in an account after 1.5 years if $4,000 is invested at 5% compounded semiannually?

Solve the problem. Round to the nearest cent as needed. 8) How many months will it take until an account will have $3,500 if $2,500 is invested now at 5% compounded monthly? Use an amortization table to solve the problem. Round to the nearest cent. 9) A bank makes a home mortgage loan of $180,000 at 7.25% amortized in equal monthly payments over 30 years. What is the total amount paid in interest when the mortgage is paid off (round to the nearest dollar)? Provide an appropriate response. 10) An investor purchased 500 shares of a stock at $19 per share. The commission she paid to buy the stock was $65 plus 0.3% of the principal amount. Six months later she sold the stock for $20.50 per share. If she paid the same rate of commission to sell the stock, what annual rate of interest (annual yield) did she earn on her initial investment (including purchase price plus commission)? Express your answer as a percentage, correct to one decimal place. 11) If you pay $5,500 for a simple interest note that will be worth $6,000 in 21 months, what annual simple interest rate will you earn? (Compute the answer to one decimal place.) Use an amortization table to solve the problem. Round to the nearest cent. 12) You have agreed to pay off a $8,000 loan in 30 monthly payments of $298.79 per month. The interest rate of the loan is 0.75% per month on the unpaid balance. What is the unpaid balance after 12 monthly payments have been made?

10)

11)

12)

13) A $7,000 debt is to be amortized in 15 equal monthly payments of $504.87 at 1.00% interest per month on the unpaid balance. What is the unpaid balance after the second payment?

13)

14) A home was purchased 14 years ago for $70,000. The home was financed by paying a 20% down payment and signing a 25 year mortgage at 8.5% compounded monthly on the unpaid balance. The market value is now $100,000. The owner wishes to sell the house. How much equity (to the nearest dollar) does the owner have in the house after making 168 monthly payments?

14)

Provide an appropriate response. 15) Find the amount due on a loan of $8,500 at 7.5% simple interest at the end of 4 years.

15)

16) An investment company pays 7% compounded quarterly. What is the effective rate? (Compute the answer to two decimal places).

16)

17) If an investor buys a 39 -week T-bill with a maturity value of $25,000 for $23,543 what annual interest rate (annual yield) will the investor earn? (Express your answer as a percentage, correct to one decimal place.)

17)

Solve the problem. 18) You deposit $130 each month into a savings account that pays 5.5% compounded monthly. How much interest will you have earned after 8 years?

18)

19) An ordinary annuity has a value of $1,333.85 at the end of 4 years when $150 is deposited every 6 months into an account earning 6% compounded semiannually. How much interest has been earned? Use an amortization table to solve the problem. Round to the nearest cent. 20) You have purchased a new house and have a mortgage for $90,000 at 6% compounded monthly. The loan is amortized over 20 years in equal monthly payments of $644.79. Find the total amount paid in interest when the mortgage is paid off. Provide an appropriate response. 21) A bank account starts with $1,000 in it. Interest is paid at 6% annual interest, compounded monthly. Graph the exact function for the amount in the account over the first 12 months. Use vertical scale [900, 1,200]. Solve the problem. 22) What is the future value of an ordinary annuity at the end of 3 years if $200 is deposited each quarter into an account earning 6% compounded quarterly? Solve the problem. Round to the nearest cent as needed. 23) A child receives a $10,000 gift toward a college education from her grandparents on her first birthday. How much money will it be worth in 17 years if it is invested at 8.25% compounded quarterly? Round your answer to the nearest cent.

19)

20)

21)

22)

23)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. Round to the nearest cent. 24) Joe wants to start an SEP-IRA that will have $460,000 in it when he retires in 15 years. How much should he invest semiannually in his IRA to do this if the interest is 15% compounded semiannually? A) $44.35.77 B) $17,612.13 C) $4448.77 D) $4790.74

24)

Solve the problem. Assume that simple interest is being calculated in each case. Round your answer to the nearest cent. 25) John Lee's savings account has a balance of $ 3965. After 27 months, what will the amount of 25) interest be at 0.3 % per year? A) $11.90 B) $26.76 C) $38.90 D) $146.85 Solve the problem. Round to the nearest cent as needed. 26) Brandon's savings account has a balance of $4069. After 4 years what will the amount of interest be at 5% compounded quarterly? A) $885.73 B) $101.73 C) $894.73 D) $899.73 Use I = Prt for simple interest to find the indicated quantity. 27) P = $22,660; r = 13%; 2.4 years. Find I. A) $7069.92 B) $15,590.08

27) C) $29,729.92

Find the payment necessary to amortize the loan. 28) $2200; 12% compounded quarterly; 8 quarterly payments A) $442.87 B) $313.40 C) $282.23

26)

D) $2945.80

28) D) $313.64

Solve the problem. Round to the nearest cent. 29) Larry wants to start an IRA that will have $ 400,000 in it when he retires in 22 years. How much should he invest semiannually in his IRA to do this if the interest is 9% compounded semiannually? A) $2889.93 B) $3032.28 C) $3019.28 D) $11,018.26 Use the formula A = P(1 + rt) to find the indicated quantity. 30) Allan borrowed $ 5500 from his father to buy a car. He repaid him after 8 months with interest of 7% per year. Find the total amount he repaid. A) $5724.58 B) $5756.67 C) $256.67 D) $5885.00 Solve the problem. Round dollar amounts to the nearest cent. Use 360 days. 31) Polson Associates bought a new computer system. To pay for the system, they borrowed $41,950 from the bank at 8 3 % interest for 130 days. Find the simple interest. 4 A) $1315.31

B) $3670.63

C) $1307.35

29)

30)

31)

D) $1325.50

Solve the problem. 32) Sammy borrowed $10,000 to purchase a new car at an annual interest rate of 11%. She is to pay it back in equal monthly payments over a 5-year period. How much total interest will be paid over the period of the loan? Round to the nearest dollar. A) $1435 B) $3630 C) $3045 D) $92

32)

What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest hundredth of a percent. 33) 6% compounded quarterly 33) A) 6.00% B) 6.18% C) 6.14% D) 6.09% Use the formula A = P(1 + rt) to find the indicated quantity. 34) P = $7996; r = 6%; t = 10 months. Find A. A) $8475.76 B) $6663.33

34) C) $399.80

D) $8395.80

Solve the problem. Round to the nearest cent as needed. 35) Samantha's savings account has a balance of $4643. After 25 years, what will the amount of interest be at 6% compounded annually? A) $15,289.16 B) $15,284.16 C) $15,275.16 D) $2785.80 Use I = Prt for simple interest to find the indicated quantity. 36) P = $13,500; t = 4 months; I = $517.50. Find r. A) 11.5% B) 7.7%

36) C) 3.8%

D) 11.7%

Find the monthly house payment necessary to amortize the following loan. 37) In order to purchase a home, a family borrows $267,000 at 10.8% for 15 yr. What is their monthly payment? Round the answer to the nearest cent. A) $3394.52 B) $2403.00 C) $3001.27 D) $19,108.39 Find the compound interest earned. Round to the nearest cent. 38) $700 at 8% compounded semiannually for 4 years A) $252.34 B) $118.90 C) $595.65 4

35)

37)

38) D) $258.00

Solve the problem. Round to the nearest cent as needed. 39) An actuary for a pension fund need to have $14.6 million grow to $22 million in 6 years. What interest rate compounded annually does he need for this investment to growth as specified. Round your answer to the nearest hundredth of a percent. A) 0.07% B) 7% C) 7.07% D) 7.7% 40) Cara knows that she will need to buy a new car in 3 years. The car will cost $15,000 by then. How much should she invest now at 12%, compounded quarterly, so that she will have enough to buy a new car? A) $9532.77 B) $12.594.29 C) $11,957.91 D) $10,520.70 Solve the problem. 41) The State Employees' Credit Union offers a 1-year certificate of deposit with an APY (or effective rate) of 5.5%. If interest is compounded quarterly, find the actual interest rate. Round to the nearest tenth of a percent. A) 5.6% B) 6.4% C) 5.1% D) 5.4% Find the rate of interest required to achieve the conditions set forth. 42) A = $ 32,000 P = $8,000 t = 20 years compounded quarterly A) 6.9919% B) 3.5489% C) 3.4959% 43) A = $ 32,000 P = $8,000 t = 20 years compounded annually A) 5.6467%

39)

40)

41)

42)

D) 5.5310% 43)

B) 7.1773%

C) 3.5265%

D) 8%

Use the future value formula to find the indicated value. Round to three decimal places. 44) n = 13; i = 0.04; PMT = $1000; FV = ? A) $16,627.838 B) $18,292.371 C) $41,627.144 D) $15,026.247 Find i (the rate per period) and n (the number of periods) for the annuity. 45) Semiannual deposits of $400 are made for 10 years into an annuity that pays 7% compounded semiannually. A) i = 0.35; n = 10 B) i = 0.07; n = 10 C) i = 0.035; n = 20 D) i = 0.0175; n = 40

44)

45)

Convert the given interest rate to decimal form if it is given as a percentage, and to a percentage if it is given in decimal form. 46) 0.05% to decimal 46) A) 0.0005 B) 5.0 C) 0.05 D) 0.5 Find the present value of the ordinary annuity. 47) Payments of $65 made quarterly for 10 years at 8% compounded quarterly A) $1789.81 B) $1778.11 C) $638.18

47) D) $1748.67

Find the future value of the ordinary annuity. Interest is compounded annually, unless otherwise indicated. 48) PMT = $7,500, i = 7% interest compounded semiannually for 4 years 48) A) $67,887.65 B) $131,254.61 C) $58,345.56 D) $282,173.37 Solve the problem. Round to the nearest cent as needed. 49) The bacteria in a 11-liter container double every 3 minutes. After 55 minutes the container is full. How long did it take to fill a quarter of the container? A) 13.8 min B) 41.3 min C) 49 min D) 27.5 min 50) Cheraw Auto Repair believes that it will need new equipment in 10 years. The equipment will cost $26,000. What lump sum should be invested today at 8% compounded semiannually, to yield $26,000? A) $21,097.18 B) $11,866.06 C) $17,637.61 D) $17,462.98

49)

50)

Convert the given interest rate to decimal form if it is given as a percentage, and to a percentage if it is given in decimal form. 51) 0.25 to percent 51) A) 0.0025% B) 2.5% C) 2.25% D) 25% Find i (the rate per period) and n (the number of periods) for the annuity. 52) Monthly deposits of $1600 are made for 6 years into an annuity that pays 9% compounded monthly. A) i = 0.75; n = 12 B) i = 0.0075; n = 72 C) i = 0.09; n = 72 D) i = 0.75; n = 6 Solve the problem. Assume no new purchases are made with the credit card. 53) The annual interest rate on a credit card is 16.99 %. If a payment of $100.00 is made each month, how long will it take to pay off an unpaid balance of $1746.42? A) 15 months B) 20 months C) 21 months D) 23 months

52)

53)

Solve for the missing value. Round to four decimal places. 54) n = 33; i = 0.05 ; PMT = $1; PV = ? A) $23.9975 B) $16.1929

54) C) $16.0025

D) $15.8027

Find the monthly house payment necessary to amortize the following loan. 55) In order to purchase a home, a family borrows $70,000 at 12% for 15 years. What is the monthly payment? A) $902.99 B) $840.12 C) $46.67 D) $700.00 Solve the problem. Round to the nearest cent. 56) Cara needs $9,000 in 6 years. What amount can she deposit at the end of each quarter at 6% interest compounded quarterly so she will have her $9,000? A) $299.37 B) $330.58 C) $1290.26 D) $314.32 Use the formula A = P(1 + rt) to find the indicated quantity. 57) A = $16,400; r = 10%; 90 days. Find P. A) $16,000 B) $16,400

55)

56)

57) C) $16,810

D) $400.00

Solve the problem. 58) At the end of every 3 months, Judy deposits $100 into an account that pays 6% compounded quarterly. After 4 years, she puts the accumulated amount into a certificate of deposit paying 7.5% compounded semiannually for 1 year. When this certificate matures, how much will Judy have accumulated? A) $2072.31 B) $1823.20 C) $1920.96 D) $1930.25

58)

Solve the problem. Assume that the minimum payment on a credit card is the greater of $27 or 3% of the unpaid balance. 59) If the annual interest rate is 15.99 %, find the difference between the minimum payment and the 59) interest owed on an unpaid balance of $728.97 that is 1 month overdue. A) $14.29 B) $12.22 C) $17.29 D) $12.76 Solve the problem. Round to the nearest cent as needed. 60) The monthly payments on a $79,000 loan at 14% annual interest are $982.76. How much of the first monthly payment will go toward interest? A) $921.67 B) $845.17 C) $1106.00 D) $137.59 Find i (the rate per period) and n (the number of periods) for the loan at the given annual rate. 61) Quarterly Payments of $3000 are made for 4 years to repay a loan at 10.2% compounded quarterly. A) i = 0.006375; n = 4 B) i = 0.102; n = 16 C) i = 0.0255; n = 4 D) i = 0.0255; n = 16 Find the payment necessary to amortize the loan. 62) $12,200 ; 12% compounded monthly; 48 monthly payments A) $316.16 B) $1470.38 C) $321.50

D) $321.27

63)

64) C) $22.50

Find the compound amount for the deposit. Round to the nearest cent. 65) $15,000 at 10% compounded semiannually for 10 years A) $39,799.47 B) $24,433.42 C) $30.000.00 66) $200 at 7% compounded quarterly for 5 years A) $270.00 B) $280.51

61)

62)

Solve the problem. Round dollar amounts to the nearest cent. Use 360 days. 63) What is the purchase price of a 26 -week T-bill with a maturity value of $1000 that earns an annual interest rate of 5.25%? A) $1000 B) $25.58 C) $1025.58 D) $974.42 Use I = Prt for simple interest to find the indicated quantity. 64) I = $750; r = 6%; t = 6 months. Find P. A) $24,250 B) $25,750

60)

D) $25,000

65) D) $38,906.14 66)

C) $218.12

Make the indicated conversion. Assume a 360-day year as needed. 67) 8 months to simplified fraction of a year 1 66 2 A) B) C) 45 100 3

D) $282.96

67) D)

8 12

Solve the problem. Round to the nearest cent as needed. 68) The rabbit population in a forest area grows at the rate of 7% monthly. If there are 180 rabbits in July, find how many rabbits (rounded to the nearest whole number) should be expected by next July. Use y = 180(2.7) 0.07t A) 415

B) 402

C) 408

Make the indicated conversion. Assume a 360-day year as needed. 69) 150 days to a simplified fraction of year 15 5 15 A) B) C) 12 12 365

D) 428

69) D)

5 36

Use an amortization table to solve the problem. Round to the nearest cent. 70) The monthly payments on a $73,000 loan at 13% annual interest are $807.38. How much of the first monthly payment will go toward the principal? A) $16.55 B) $790.83 C) $702.42 D) $104.96 Solve for the missing value. Round to four decimal places. 71) n = 30; i = 0.03; PMT = $100; PV = ? A) $2000.0441 B) $1918.1585

68)

70)

71) C) $1960.0441

D) $4706.6287

Use an amortization table to solve the problem. Round to the nearest cent. 72) The monthly payments on a $76,000 loan at 12% annual interest are $836.76. How much of the first monthly payment will go toward interest? A) $760.00 B) $912.00 C) $100.41 D) $736.35 Find the periodic payment that will render the sum. 73) FV = $ 42,000 , interest is 8% compounded monthly, payments made at the end of each month for 3 years A) $12,433.54 B) $3373.51 C) $1036.13 D) $2213.19

72)

73)

Use the average daily balance method to compute the amount of interest that will be charged at the end of the billing cycle. Use a 365-day year. 74) Month: February (28 days) 74) Previous month's balance: $ 1250 Interest rate: 22% Date Transaction February 3 made payment of $ 320 February 12 purchase of $321 February 21 purchase of $43 February 22 purchase of $67 A) $19.86 B) $16.09 C) $18.35 D) $8.38 Use the future value formula to find the indicated value. Round to three decimal places. 75) n = 15; i = 0.04; PMT = $1; FV = ? A) $21.825 B) $20.024 C) $45.024 D) $18.292

75)

Solve the problem. Round to the nearest cent as needed. 76) How long will it take for $8400 to grow to $14,600 at an interest rate of 9.4% if the interest is compounded continuously? Round the number of years to the nearest hundredth. A) 5.88 years B) 0.06 year C) 0.59 year D) 58.81 years Find the rate of interest required to achieve the conditions set forth. 77) If Jay bought a lot for $8,000 and sold it 15 years later for $24,000, what was her percentage rate of return on this investment if it was compounded annually? A) 7.5990% B) 9.6825% C) 3.7995% D) 8.7104% Solve the problem. 78) If $300,000 is to be saved over 25 years, how much should be deposited monthly if the investment earns 8% interest compounded monthly? A) $260.87 B) $180.48 C) $216.62 D) $315.45

76)

77)

78)

Use the average daily balance method to compute the amount of interest that will be charged at the end of the billing cycle. Use a 365-day year. 79) Month: May (31 days) 79) Previous month's balance: $ 950 Interest rate: 18% Date Transaction May 3 made payment of $ 360 May 12 purchase of $110 May 21 purchase of $89 May 29 made payment of $ 68 A) $8.40 B) $11.82 C) $7.55 D) $10.84 Solve the problem. Round to the nearest cent as needed. 80) The monthly payments on a $73,000 loan at 13% annual interest are $807.38. How much of the first monthly payment will go toward the principal? A) $790.83 B) $702.42 C) $104.96 D) $16.55

80)

Use the average daily balance method to compute the amount of interest that will be charged at the end of the billing cycle. Use a 365-day year. 81) Month: July (31 days) 81) Previous month's balance: $ 840 Interest rate: 16% Date Transaction July 5 made payment of $ 201 July 15 purchase of $41 July 20 purchase of $86 July 27 purchase of $68 A) $8.47 B) $9.94 C) $8.75 D) $8.54 Solve the problem. Round dollar amounts to the nearest cent. Use 360 days. 82) HarbourTown Marina purchased four boat lifts for raising and lowering large boats into the water. The boat lifts cost $61,300 each. They borrowed the money from the bank for 240 days at 10%. Find the maturity value. A) $261,546.67 B) $67,430.00 C) $65,386.67 D) $269,720.00

82)

Use I = Prt for simple interest to find the indicated quantity. 83) P = $3000; t = 90 days; I = $105. Find r. (Use 360 days in a year.) A) 3.5% B) 14.2% C) 9.7%

83) D) 14.0%

Solve the problem. Assume that the minimum payment on a credit card is the greater of $27 or 3% of the unpaid balance. 84) Find the minimum payment on an unpaid balance of $769.52. 84) A) $230.86 B) $28.09 C) $27.00 D) $23.09 Use the average daily balance method to compute the amount of interest that will be charged at the end of the billing cycle. Use a 365-day year. 85) Month: April (30 days) 85) Previous month's balance: $ 960 Interest rate: 21% Date Transaction April 3 purchase of $106 April 12 made payment of $ 340 April 21 purchase of $95 April 29 made payment of $ 69 A) $15.03 B) $11.69 C) $17.62 D) $14.16 Find the compound amount for the deposit. Round to the nearest cent. 86) $1100 at 3% compounded quarterly for 2 years A) $1116.56 B) $1167.76 C) $1166.00

86) D) $1166.99

Solve the problem. Assume that the minimum payment on a credit card is the greater of $27 or 3% of the unpaid balance. 87) Find the minimum payment on an unpaid balance of $1456.38 . 87) A) $43.69 B) $48.69 C) $27.00 D) $436.91 Solve the problem. 88) Jennifer invested $7000 in her savings account for 4 years. When she withdrew it, she had $8792.60 . Interest was compounded continuously. What was the interest rate on the account? Round to the nearest tenth of a percent. A) 5.6 % B) 5.85 % C) 5.7% D) 5.8 % Solve the problem. Round to the nearest cent as needed. 89) A bank has $750,000 to lend for 7 months. It can lend it to a local contractor at a simple interest rate of 12%, or it can lend it to a small business that will pay 12% compounded monthly. If the bank wants to maximize its interest earned, who should receive the loan (contractor or business) and what is the additional interest earned? A) Contractor; $1601.51 B) Business; $1601.51 C) Contractor; $52,500 D) Business; $54,101.51 Find i (the rate per period) and n (the number of periods) for the loan at the given annual rate. 90) Annual payments of $378.80 are made for 8 years to repay a loan at 6.4% compounded annually. A) i = 0.064; n = 8 B) i = 0.00533; n = 12 C) i = 0.008; n = 8 D) i = 0.064; n = 16

88)

89)

90)

Convert the given interest rate to decimal form if it is given as a percentage, and to a percentage if it is given in decimal form. 91) 11.6% to decimal 91) A) 0.00116 B) 116 C) 0.116 D) 11.6 What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest hundredth of a percent. 92) 5% compounded semiannually 92) A) 5.06% B) 5.13% C) 5.09% D) 5.00% Find the compound interest earned. Round to the nearest cent. 93) $600 at 5% compounded quarterly for 1 1 years 2 A) $95.82

B) $46.43

C) $95.45

94) $14,000 at 5% compounded annually for 3 years A) $2100.00 B) $701.50

93) D) $204.06 94)

C) $2206.75

D) $1435.00

Find the amount that will be accumulated in the account under the given conditions. 95) The principal $15,400 is accumulated with simple interest of 16% for 5 years. A) $20,212.50 B) $15,892.80 C) $27,720 D) $12,320

95)

What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest hundredth of a percent. 96) 3.5% compounded continuously. 96) A) 3.53% B) 3.56% C) 3.50% D) 3.55% Convert the given interest rate to decimal form if it is given as a percentage, and to a percentage if it is given in decimal form. 97) 0.05 to percent 97) A) 50% B) 0.0005% C) 0.05% D) 5% Solve the problem. 98) How long will it take for $ 9500 to grow to $ 35,400 at an interest rate of 10.1% if the interest is compounded continuously? Round the number of years to the nearest hundredth. A) 1302.4 yr B) 1.3 yr C) 0.13 yr D) 13.02 yr

98)

Answer Key Testname: CHAP 03_14E

1) 21.1% 2) 62 months, or 5 years, 2 months 3) $20,978 4) $7,332.17 5) $100.45 6) $68.32 7) $4,307.56 8) 81 months or 6 years, 9 months 9) $262,051.20 10) 11.7% 11) 5.2% 12) $5,013.45 13) $6,125.91 14) $61,414 15) $11,050 16) 7.19% 17) 8.3% 18) $3,152.54 19) $133.85 20) $64,749.60 21)

[0, 12] by [900, 1,200] 22) $2,608.24 23) $40,077.60 24) C 25) B 26) C 27) A 28) B 29) B 30) B 31) D 32) C 33) C 34) D 35) B 36) A 37) C 38) D 39) C 12

Answer Key Testname: CHAP 03_14E

40) D 41) D 42) A 43) B 44) A 45) C 46) A 47) B 48) A 49) C 50) B 51) D 52) B 53) C 54) C 55) B 56) D 57) A 58) D 59) C 60) A 61) D 62) D 63) D 64) D 65) A 66) D 67) C 68) A 69) B 70) A 71) C 72) A 73) C 74) A 75) B 76) A 77) A 78) D 79) D 80) D 81) B 82) A 83) D 84) C 85) A 86) B 87) A 88) C 13

Answer Key Testname: CHAP 03_14E

89) B 90) A 91) C 92) A 93) B 94) C 95) C 96) B 97) D 98) D

CHAPTER 4

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. 1) Which of the following matrices has an inverse? 0 -1 A) 3 5 B) 0 4 0 -2 -1 3

1) C) 3 -2 1 4 0 7

D) -2 3 4 1

Provide an appropriate response. 2) Determine which of the following matrix equations represents the solution to the system: 2x1 + x2 = 2 . 5x1 + 3x2 = 13 A)

B) x1 = 2 x2 16

x1 = 3 -1 x2 -5 2

3 -1 -5 2

2 13

D) x1 = 2 1 x2 5 3

x1 = 2 x2 13

2 13

-2 -1 -5 -3

3) Only one of the following augmented matrices of a linear system is in a reduced form. Choose the matrix that is in reduced form. A) B) C) D) 1 4 0 4 1 0 -2 2 1 0 -2 0 1 -2 0 0 1 -4 0 0 1 3 0 0 0 1 0 -3 0 0 0 0 0 1 -1

4) Given A = -1 2 -3 , B = 7 , C = 9 -1 5 , and D = 1 0 , determine which of the -4 5 6 -2 -2 6 following products is NOT defined. A) DA B) AD C) DB D) BC

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Perform the operation, if possible. 5) Let A = 0 -2 1 4 -1 0

and B =

1 -2 0 1 . Find AB. -2 -1

Provide an appropriate response. -3x1 + x2 - x3 = 2 -3 1 -1 -1 1 -1 1 = 0 6) Use -3 1 0 = 3 -2 3 to solve -3x1 + x2 1 0 1 -1 1 0 x1 + x3 = -3

Use the given encoding matrix A to solve the problem. 7) The following message was encoded with matrix 1 1 . Decode this message. 2 3

28 64 32 91 30 65 24 60 38 99 42 99 35 82 36 81 46 119 13 31 23 51 Solve the problem. 8) A retail company offers, through two different stores in a city, three models, A, B, and C, of a particular brand of camping stove. The inventory of each model on hand in each store is summarized in matrix M. Wholesale (W) and retail (R) prices of each model are summarized in matrix M. Find the product MN and label its columns and rows appropriately. What is the wholesale value of the inventory in Store 1? A B C W R $60 $90 A M = 2 0 1 Store 1 N = $120 $150 B 3 3 0 Store 2 $40 $50 C

Provide an appropriate response. 9) Solve the matrix equation 2 1 5 3 matrix.

10) Given matrices M =

x1 + 1 = 3 by using the inverse of the coefficient x2 5 18

-1 2 0 x 2 2 3 1 , X = y , A = 4 , M-1 = -4 0 -2 z 1

-1

2 3

1 3

2 -

1 3 6 , and B = 0 6 7

4 3 6

10)

solve the matrix equations MX = A and MX = B. Solve the problem. 11) Suppose that the supply and demand equations for a logo sweat shirt in a particular week are p = 55 - 0.10q, for the demand equation; and p = 0.20q + 25 , for the supply equation. Find the equilibrium price and quantity. 12) A company that manufactures laser printers for computers has monthly fixed costs of $177,000 and variable costs of $650 per unit produced. The company sells the printers for $1,250 per unit. How many printers must be sold each month for the company to break even? Solve the system of equations by elimination. 13) 8x - 4y = 10 12x - 6y = -15

11)

12)

13)

Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM. 14) Labor and material costs for manufacturing each of three types of products M, N, and P are given in the table: Product M N P Labor $50 $40 $50 Materials $60 $40 $70

14)

The weekly allocation for labor is $50,000 and for materials is $80,000. There are to be 3 times as many units of product M manufactured as units of product P. How many of each type of product would be manufactured each week to use exactly each of the weekly allocations? Set up a system of linear equations, letting x1, x2, and x3 be the number of units of products M, N, and P, respectively, manufactured in one week. Solve the problem. 15) Given the technology matrix M and the final demand matrix D stated below, find (I - M) -1 and find the output matrix X. M = 0.3 0.3 0.2 0.2

15)

D = 70 30

Provide an appropriate response. 16) Use Gauss-Jordan elimination, without introducing fractions, to find the inverse of -4 -2 -5 0 1 0 . 1 0 1 17) Use a graphing utility and augmented matrix methods to solve the system: 1.8 x1 - 12.17x2 = 33

16)

17)

-3.75 x1 + 5.73x2 = 7 Express your answer accurate to three decimal places. Perform the operation, if possible. 18) Let A = x 5 and B = 4 3 . Find BA. 2 y 8 7

18)

Provide an appropriate response. 19) Solve the linear system corresponding to the following augmented matrix:

19)

1 4 0 5 0 0 1 6 0 0 0 0 Solve the system of equations by graphing. 20) Use a graphing utility to solve the system y = 2x + 7 . Give the answer to three decimal y = -5x + 1 places.

20)

Perform the operation, if possible. 21) If a and b are nonzero real numbers and A =

ab , find A2. -ab -a2

Provide an appropriate response. 22) Given matrix A: 5 -3 1 7 0 -2 A= 1 -9 2

21)

22)

What is the size of A? Find a32 and a11. 23) Use the matrix method on a graphing calculator to solve the system

23)

4x1 + 6x2 - x3 + 4x4 = 81 x1 - 3x2 + 2x3 + 10x4 = 95 -3x1 + 7x2 - 4x3 x1 + x2 + x3 +

x4 = 12 x4 = 0

Carry values to two decimal places. 24) Solve the matrix equation -3 4 x = 25 1 -2 y -11 by using the inverse of the coefficient matrix. Also, solve the system if the constants 25 and -11 are replaced by 1 and 3, respectively. Solve the problem. 25) A large oil company produces three grades of gasoline: regular, unleaded, and super-unleaded. To produce these gasolines, equipment is used which requires as input certain amounts of each of the three grades of gasoline. To produce a dollar's worth of regular requires inputs of $0.14 worth of regular, $0.18 worth of unleaded, and $0.17 worth of super-unleaded. To produce a dollar's worth of unleaded requires inputs of $0.14 worth of regular, $0.15 worth of unleaded, and $0.13 worth of super-unleaded. To produce a dollar's worth of super-unleaded requires inputs of $0.15 worth of regular, $0.17 worth of unleaded, and $0.11 worth of super-unleaded. In addition, the oil company has final demands for each of the different grades of gasoline. Find the technology matrix that would be used in determining the total output of each grade of gasoline.

24)

25)

26) A supermarket chain sells oranges, apples, peaches, and bananas in three stores located throughout a large metropolitan area. The average number of pounds sold per day in each store is summarized in matrix M. "In season" and "out of season" prices, per pound, of each fruit are given in matrix N. What is the total, for the three stores, of "in season" daily revenue for the four fruits? The "out of season" peach sales represent what percentage of the daily total "out of season" revenues for store 3?

26)

Fruit O A P B 60 80 60 55 Store 1 M = 95 80 65 75 Store 2 85 85 70 95 Store 3 "In season" $3.00 N = $5.00 $5.00 $0.40

"Out of season" $7.00 O $9.50 A $5.50 P $0.60 B

27) A hospital dietitian wants to insure that a certain meal consisting of rice, broccoli, and fish contains exactly 26,800 units of vitamin A, 840 units of vitamin E, and 11,160 units of vitamin C. One ounce of rice contains 400 units of vitamin A, 20 units of vitamin E, and 180 units of vitamin C. One ounce of broccoli contains 800 units of vitamin A, 60 units of vitamin E, and 540 units of vitamin C. And one ounce of fish contains 2,400 units of vitamin A, 40 units of vitamin E, and 810 units of vitamin C. How many ounces of each food should this meal include? Set up a system of linear equations and solve using Gauss-Jordan elimination.

27)

28) A chain of amusement parks pays experienced workers $240 per week and inexperienced workers $220 per week. The total number of workers and total weekly wages at three different parks are given in the table. How many experienced workers does each park employ? Set up a system of linear equations and solve using matrix inverse methods.

28)

Number of workers Total weekly wages

Park 1 120 28,400

Park 2 120 27,200

Park 3 120 28,000

29) A trucking firm wants to purchase 10 trucks that will provide exactly 28 tons of additional shipping capacity. A model A truck holds 2 tons, a model B truck holds 3 tons, and a model C truck holds 5 tons. How many trucks of each model should the company purchase to provide the additional shipping capacity? Set up a system of linear equations and solve using Gauss -Jordan elimination. There may be more than one solution.

29)

30) An economy is based on two sectors, agriculture and manufacturing. Production of a dollar's worth of agriculture requires an input of $0.40 from agriculture and $0.10 from manufacturing. Production of a dollar's worth of manufacturing requires an input of $0.20 from agriculture and $0.30 from manufacturing. Find the output for each sector that is needed to satisfy a final demand of $16 billion for agriculture and $32 billion for manufacturing. Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM. 31) In producing three types of bricks: face bricks, common bricks, and refractory bricks, a factory incurs labor, material, and utility costs. To produce one pallet of face bricks, the labor, material, and utility costs are $50, $75, and $35, respectively. To produce one pallet of common bricks, the labor, material, and utility costs are $50, $60, and $30, respectively, while the corresponding costs for refractory bricks are $75, $100, and $45. In a certain month the company has allocated $12,000 for labor costs, $14,500 for material costs and $6,000 for utility costs. How many pallets of each type of brick should be produced in that month to exactly utilize these allocations? Set up a system of linear equations, letting x, y, and z be the number of pallets of face, common, and refractory bricks, respectively, that must be produced in that month. 32) A paper company produces high, medium, and low grade paper. The number of tons of each grade that is produced from one ton of pulp depends on the source of that pulp. The following table lists three sources and the amount of each grade of paper that can be made for one ton of pulp from each source.

Brazilian Pulp Domestic Pulp Recycled Pulp

(Number of Tons) Medium High Grade Grade 0.6 0.3 0.5 0.3 0.3 0.4

30)

31)

32)

Low grade 0.1 0.2 0.3

The paper company has orders for 11 tons of high grade, 15 tons of medium grade, and 14 tons of low grade paper. How many tons of each type of pulp should be used to fill these orders exactly? Set up a system of linear equations, letting x, y, and z be the number of tons of Brazilian pulp, domestic pulp, and recycled pulp, respectively, needed to fill the orders. 33) If $9,000 is to be invested, part at 13% and the rest at 8% simple interest, how much should be invested at each rate so that the total annual return will be the same as $9,000 invested at 9%? Set up a system of linear equations, letting x1 be the amount invested at

33)

13% and x2 be the amount invested at 8%. Provide an appropriate response. 34) Solve the linear system corresponding to the following augmented matrix: 1 0 2 -5 0 1 -4 2 0 0 1 1

34)

35) Use Gauss-Jordan elimination to find the inverse of 1 1 . 3 4 -1 36) Use -7 6 = -1 -6 to solve 1 -1 -1 -7

35)

-7x1 + 6x2 = -5 . x1 - x2 = -5

36)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the system of equations by graphing. 37) 2x + y = 5 3x + y = 6 10

37)

-10

-5

10 x

5 -5 -10

A) (3, 1)

B) (-1, -3)

C) (1, 3)

D) no solution

Use the given encoding matrix A to solve the problem. 38) Use the given message to construct the code matrix by assigning numbers to the letters and symbols. Use the numerical assignment a = 1, b = 2, . . . , z = 26, space = 30, period = 40, and apostrophe = 60. Message: CALL ME TOMORROW. Encoding matrix A = 1 0 1 1 A) B)

38)

D) 3 12 5 15 18 23

1 30 30 13 18 40

12 13 20 15 15 30

3 12 30 5 18 13 19 15 40

1 12 14 30 15 15 19 23 30

Solve the equation for the indicated variable. Assume that the dimensions are such that matrix multiplication and addition are possible and that inverses exist when needed. 39) Solve for C: CZ = Y 39) -1 -1 -1 A) C = Z Y B) C = Y/Z C) C = YZ D) C = ZY

Solve the system of equations by elimination. 40) 2x + y = -1 4x - y = 7 A) (-3, 1) C) infinitely many solutions

40) B) (1, -3) D) no solution

Solve the system as matrix equations using inverses. 41) -5x1 + 3 x2 = 8 3x1 - 6 x2 = -30 A) (-6, -2)

41)

B) (6, 2)

C) (-2, -6)

Solve the system of equations by elimination. 42) x + y = 6 x- y=6 A) (6, 0) B) (0, -6)

D) (2, 6)

42) C) (0, 6)

Solve the system of equations by substitution. 43) x - 4y = -2 y=1 A) (1, 2) B) (-2, 1)

D) (-6, 0)

43) C) (-6, 1)

Find the values of a, b, c, and d that make the matrix equation true. 44) a b + 0 5 = 3 7 c d -4 7 2 -1 3 12 A) B) 0 2 C) 3 2 -2 6 6 8 6 -8

D) (2, 1)

44) D) -3 -2 -6 8

Solve the problem. 45) A textbook economy has only two industries, the electric company and the gas company. Each dollar's worth of the electric company's output requires 0.20 of its own output and 0.4 of the gas company's output. Each dollar's worth of the gas company's output requires 0.50 of its own output and 0.7 of the electric company's output. What should the production of electricity and gas be (in dollars) if there is a $16 M demand for electricity and a $7 M demand for gas? A) Electricity: $107.5 M; Gas: $100 M B) Electricity: $97.5 M; Gas: $103 M C) Electricity: $125 M; Gas: $92.5 M D) Electricity: $115 M; Gas: $103.5 M Perform the indicated operations given the matrices. 46) Let A = -4 2 and B = 1 0 ; 2A + 3 B A) -1 2 B) -5 4

45)

46) C)

-7 4

-8 4

Find the matrix product mentally, without the use of a calculator or pencil-and-paper calculations. 47) 1 0 2 3 0 1 5 1 1 1 2 3 A) 2 3 B) 1 1 C) 3 2 D) 1 5 1 1 1 1 5 1 5

47)

Perform the operation, if possible. 48) - 1 0 - -1 3 3 1 3 1 -2 3 A) 6 2

48) B) 0 -3 0 0

D) 0 0

C) -3

3 0

Solve the problem. 49) A chemistry department wants to make 3 liters of a 17.5% basic solution by mixing a 20% solution with a 15% solution. How many liters of each type of basic solution should be used to produce the 17.5% solution? A) 0.5 liter of 15% solution, 2.5 liters of 20% solution B) 1 liter of 15% solution, 2 liters of 20% solution C) 1.5 liters of 15% solution, 1.5 liters of 20% solution D) 2 liters of 15% solution, 1 liter of 20% solution Perform the operation, if possible. 50) -9 1 + 6 2 2 5 2 -3 -3 -7 A) -7 -12

50) B)

3 4

3 2

C) -3 4

3 2

D) -3 4

-3 2

Solve the problem. 51) A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the number of units of each ingredient in each type of candy in one batch. Matrix B gives the cost of each ingredient (dollars per unit) from suppliers X and Y. What is the cost of 100 batches from supplier X?

A) $6600

49)

B) $3300

C) $4800

51)

D) $7800

Perform the operation, if possible. -6 -2 9 52) Let A = -1 5 1 and B = -5 -7 -3 . Find AB. 6 -8 2

52)

A) 13 41 22

B) -13 -41 -22

-13 C) -41 -22

6 -10 9 5 -35 -3 -6 -40 2

Perform the indicated operations given the matrices. 1 -1 53) Let C = -3 and D = 3 ; C - 4D 2 -2 -3 -5 A) B) 9 15 -6 -10

53)

5 -6 4

5 -15 10

Provide an appropriate response. 54) Write the augmented matrix for the system. 6 x1 + 4x2 = 30

54)

8x2 = 72

A) 8 0 6 4

72 4

B) 30 4 72 0

6 8

C) 6 4 30 8 72 0

D) 6 4 30 0 8 72

Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM. 55) Hurst's Feed & Seed sold to one customer 5 bushels of wheat, 2 of corn, and 3 of rye, for $31.00. To another customer he sold 2 bushels of wheat, 3 of corn, and 5 of rye, for $27.60. To a third customer he sold 3 bushels of wheat, 5 of corn, and 2 of rye for $32.70. What was the price per bushel for each of the different grains? Let x represent the price per bushel for wheat, y the price per bushel for corn, and z the price per bushel for rye. A) 5x + 2y + 3z = 31.00 B) 5x + 2y + 3z = 31.00 2x - 3y + 5z = 27.60 2x + 3y + 5z = 27.60 3x + 5y + 2z = 32.70 3x + 5y + 2z = 32.70 C) 5x + 2y + 3z = 31.00 D) 5x + 2y - 3z = 31.00 2x + 3y - 5z = 27.60 2x + 3y - 5z = 27.60 3x + 5y - 2z = 32.70 3x + 5y - 2z = 32.70 Solve the problem. 56) The input-output matrix for an economy is Output: Agri. Mfg. 0.18 = T Input: Agri. 0.04 Mfg. 0.02 0.22 The demand matrix is D =

55)

56)

700 1000

Find the internal consumption. A) B) 507.2 374.2 57.0 397.3

D) 161.5 108.3

274.2 307.0

Write a system of equations associated with the augmented matrix. Do not try to solve. 57) 2 0 -3 x = 1 1 2 y -5 -5 2 A) 2x + y = -3 x + 2y = 2 -5x - 5y = 2

B) 2x = -3 x+ y =2 -5x - 5y = 2

C) 2x + y + z = -3 x - 5y =2

Write the matrix equation as a system of linear equations without matrices. x1 3 3 5 -2 58) 5 0 7 x2 = 4 3 6 0 x3 2 A) 3x1 + 3 x2 + 5 x3 = 2

B) 3x1 - 3 x2 + 5 x3 = - 2

5x1 + 7x3 = 4 3x1 + 6 x2 = 2

C) 3x1 + 3 x2 + 5 x3 = - 2

D) 3x1 + 3 x2 + 5 x3 = - 2

5x1 + 7x3 = 4 3x1 + 6 x2 = 2

5x1 + 7x3 = - 4 3x1 + 6 x2 =-2

Solve the problem. 59) Daisy has a desk full of quarters and nickels. If she has a total of 23 coins with a total face value of $4.35, how many of the coins are nickels? A) 9 nickels B) 7 nickels C) 16 nickels D) 21 nickels 60) Two sectors of a textbook economy are (1) communication equipment and (2) components and accessories. In 2005 the input-output table involving these two sectors was as follows. To Equipment Components From Equipment 6,000 500 Components 24,000 30,000 Total Output 90,000 140,000 Determine the production levels necessary in these two sectors to meet a demand for $80,000 of equipment and $90,000 of components. Round to significant digits. A) Equipment: 90,000 B) Equipment: 24,000 Components: 140,000 Components: 140,000 C) Equipment: 86,000 D) Equipment: 86,000 Components: 90,000 Components: 140,000 Find the matrix product mentally, without the use of a calculator or pencil-and-paper calculations. 1 0 0 1 2 3 61) 0 1 0 4 5 6 0 0 1 7 8 9 9 8 7 1 -2 3 1 2 3 A) 6 5 4 B) -4 5 -6 C) 4 5 6 D) 1 3 2 1 7 -8 9 7 8 9

57)

58)

59)

60)

61)

Solve the system as matrix equations using inverses. 62) - 2x1 + 6 x2 = 6 3x1 + 2x2 = 13 A) (2,3)

62)

B) (-2, -3)

C) (3,2)

D) (-3 , - 2)

State whether the matrix is in reduced form or not in reduced form. 1 3 0 1 63) 0 0 1 0 0 0 0 1 A) Reduced Form B) Not Reduced Form Solve using Gauss-Jordan elimination. 64) -4x - y + 5 z = 21 -8x + 8 y - 4z = -4 9 x - 6 y + z = -3 A) (4, 9, 8) B) (4, 8, 9)

63)

64)

C) (-4, 8, 8)

D) No solution

Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM. 65) A $124,000 trust is to be invested in bonds paying 9%, CDs paying 8%, and mortgages paying 10%. The sum of the amount invested in bonds and the amount invested in CDs must equal the mortgage investment. To earn an $11,400 annual income from the investments, how much should the bank invest in each? Let x represent the amount invested in bonds, y the amount invested in CDs, and z the amount invested in mortgages. A) x + y - z = 0 B) x + y - z = 0 x + y + z = 124,000 x + y + z = 124,000 0.09x + 0.08y + 0.1z = 11,400 9x + 8y + z = 11,400 C) x + y + z = 0 D) x + y - z = 11,400 x + y - 9z = 124,000 x - y + 9z = 22 0.1x + 0.08y - 0.09z = 11,400 8x + y + z = 124,000 Perform the operation, if possible. 66) 5 4 + - 2 7 A) 3

66) B)

3 11

C) 5 - 2 4 7

D) Not defined

Identify the row operation that produces the resulting matrix. 1 0 2 67) -1 1 3 → 1 0 2 0 1 5 A) -R 1 + R 2 → R 2

65)

B) R 1 + R 2 → R 2

67) C) -R 2 + R 1 → R 2

Determine whether B is the inverse of A. 2 -1 0 1 -1 2 68) A = -1 1 -2 , B = -3 -2 4 1 0 -1 -1 1 1 A) Yes

D) R 1 + R 2 → R 1

68) B) No

Write the system as a matrix equation of the form AX = B. 69) 8 x1 + 9 x2 = 117

69)

4x1 + 6 x2 = 66 x1 A) 8 4 = 117 x 9 6 66 2 x1 C) 8 9 = 117 4 6 x2 66

x1 = 66 x2 117 x1 D) 117 9 = 8 66 6 x2 6 B) 8 9 6 4

Perform the indicated row operations on the following matrix. 1 -5 4 2 2 5 70) (-2)R 1 + R 2 → R 2 A) -2 10 -8 0 12 -3

B) 1 -5 4 0 12 -3

C) -2 10 -8 2 12 -5

70) D) 0 12 -3 2 2 5

Solve the system as matrix equations using inverses. 71) x1 + x2 + x3 = -2

71)

x1 - x2 + 3 x3 = 8 5x1 + x2 + x3 = -22 A) (4, -5, -1)

B) (5, 1, 4)

C) (-5, -1, 4)

Solve using Gauss-Jordan elimination. 72) 9x - 8y = -2 -9x + 8y = 9 9 A) 4x - y, y B) (-1, -1) 2

D) (4, -1, -5)

72) C) (-2, 9)

D) No solution

Solve the equation for the indicated variable. Assume that the dimensions are such that matrix multiplication and addition are possible and that inverses exist when needed. 73) Solve for A: AY - A = B 73) -1 -1 A) A = B(Y - I) B) A = Y B - I C) A = BY -1 + I

D) A = (Y - I) -1 B

State whether the matrix is in reduced form or not in reduced form. 1 0 -1 5 74) 0 4 1 1 0 0 11 A) Reduced Form B) Not Reduced Form

74)

Write the matrix equation as a system of linear equations without matrices. 75) 6 0 -5 x1 = 1 1 7 x2 8 -5 6 A) 6 x1

= -5

B) 6 x1 + x2 = -5

x1 + x2 = 7 8 x1 - 5 x2 = 6

x1 + 7x2 = 7 8 x1 - 5 x2 = 6

Perform the operation, if possible. 76) A = 3 0 , B = 3 -2 1 -2 1 0 4 -3 A) 9 -6 3 -6 8 -5

75)

C) 6 x1 + x2 + x3 = -5 x1 - 5 x2

. Find BA.

76) B) BA is not defined.

D) 9 -6 -6 8 3 -5

9 0

User row operations to change the matrix to reduced form. 1 -1 0 1 77) 0 4 8 4 0 0 0 0 1 0 2 0 1 -1 0 1 A) 0 1 2 1 B) 0 1 2 1 0 0 0 0 0 0 0 0 Find the inverse, if it exists, of the given matrix. 78) 5 4 6 5 A) B) -5 6 -5 -4 4 -5 -6 -5

0 4

77) 1 0 2 2 C) 0 1 2 0 0 0 0 0

1 0 2 2 D) 0 1 2 1 0 0 0 0

78) C)

D) 5 6 4 5

Solve the system of equations by elimination. 79) x - 5y = 35 5x - 4y = 28 A) (1 , - 7) B) (1, - 8)

5 -4 -6 5

79) C) (7, 0 )

D) (0, - 7)

Solve the system mentally, without the use of a calculator or pencil-and-paper calculation. Try to visualize the graphs of both lines. 80) x - 0y = 5 80) 3x + y = 7 A) x = 3, y = 7 B) x = 5; y = -8 C) x = -5; y = 8 D) x = 5; y = 4

Write the system as a matrix equation of the form AX = B. 81) 6 x1 + 4x2 = 30

81)

8x2 = 72 x1 A) 6 4 = 30 x 0 8 72 2 x1 C) 6 4 = 30 8 72 x2 0

x1 = 72 x2 4 x1 D) 30 4 = 6 72 0 x2 8 B) 8 0 6 4

Provide an appropriate response. 82) Determine the value of each variable. x+3 y+4 = 6 0 7 -1 7 k A) x = - 3 B) x = 3 y=-4 y=-4 k=-1 k=-1

82)

C) x = - 3 y= 4 k=-1

D) x = 6 y= 0 k=-1

Solve the system as matrix equations using inverses. 83) There were 340 people at a play. The admission price was $2 for adults and $1 for children. The admission receipts were $490. How many adults and how many children attended? A) 190 adults and 150 children B) 122 adults and 218 children C) 95 adults and 245 children D) 150 adults and 190 children Solve the problem. 84) The input-output matrix for an economy is Output: Agri. Mfg. Agri. 0.04 0.18 = T Input: Mfg. 0.02 0.22 The demand matrix is D =

83)

84)

700 1000

Find the production matrix X. A) B) 974.2 X= X = 1207.2 1307.0 985.0

D) X = 1074.2 1397.3

Solve using Gauss-Jordan elimination. 85) x + y + z = 1 x - y + 5 z = 23 5x + y + z = -11 A) (5, -3, -1) B) (5, -1, -3)

861.5 1108.3

85)

C) (-3, -1, 5)

D) No solution

The system cannot be solved by matrix inverse methods. Find a method that could be used and then solve the system. 86) -2x1 + 6 x2 = -4 86) 6x1 - 18 x2 = 12 A) x1 = 3t + 2, x2 = t for any real number t

B) x1 = 2t + 6, x2 = t for any real number t

C) No Solution

D) x1 = 3t + 2 for any real number t, x2 = 0

The matrix is the final matrix form for a system of two linear equations in variables x1 and x2. Write the Solution of the system. 87) 1 0 5 0 1 3 A) x1 = 5

87) B) x1 = 5

x2 = 3 C) x1 = -5

x2 = t for any real number t D) x1 = 3

x2 = -3

x2 = 5

Use the given encoding matrix A to solve the problem. 88) A message has been encoded and the matrix which the receiver gets is shown below.

88)

The encoding matrix A which was used to encode the message is: A= 0 2 3 1 Find the decoding matrix A -1, and use it to decode the message. Assume that the numerical assignment used was a = 1, b = 2, ....., z = 26, space = 30, period = 40, and apostrophe = 60. A) DRINK ENOUGH MILK C) EAT YOUR VEGETABLES Perform the operation, if possible. 3 9 -6 7 89) Let A = -9 -4 and B = 7 -4 7 -7 -8 9

B) DRINK ENOUGH COKE D) EAT YOUR BROCCOLI

. Find A + B.

89)

B) -3 16 2 -4 -1 -2

C) -3 16 -2 -8 -1 2

D) 9 2 -16 -16 15 0

-3 -4 -2 -8 -1 2

State whether the matrix is in reduced form or not in reduced form. 90) 1 -5 0 0 1 0 0 1 2 0 A) Not Reduced Form B) Reduced Form

90)

The system cannot be solved by matrix inverse methods. Find a method that could be used and then solve the system. 91) -2x1 + 6 x2 = 4 91) 6x1 - 18 x2 = 12 A) x1 = 2t + 6, x2 = t for any real number t

B) x1 = 3t + 2, x2 = t for any real number t

C) x1 = 3t + 2 for any real number t, x2 = 0

D) No Solution

Solve the system of equations by graphing. 92) 2x - y = 3 y=1 10

92)

-10

-5

10 x

-5 -10

A) (-2, -1)

B) (1, 2)

C) (2, 1)

D) no solution

Write the linear system corresponding to the reduced augmented matrix. 1 0 4 93) 0 1 0 0 0 0 A) No Solution B) x1 = 4, x2 = 0 C) x1 = 4, x2 = t for any real number t Provide an appropriate response. 94) Given the matrix B: -3 B = -1 1 What is the size of B? A) 1 × 1

93)

D) x1 = -4, x2 = 0

94)

B) 1 × 3

C) 3

Solve the system of equations by substitution. 95) x + 3 y = -21 -6x + 4y = -28 A) (1, -8) B) (7, 0)

D) 3 × 1

95) C) (0, -7)

D) no solution

Solve the system mentally, without the use of a calculator or pencil-and-paper calculation. Try to visualize the graphs of both lines. 96) x + 0y = 9 96) 0x + y = 4 A) x = 9; y = 2 B) x = 2; y = 9 C) x = 9; y = 4 D) x = 4; y = 9 2 Identify the row operation that produces the resulting matrix. 3 0 9 97) -2 4 8 → 3 0 9 0 2 7

97)

1 1 R + R → R1 2 1 3 2

1 1 R + R → R2 3 1 2 2

D) R 1 +

Provide an appropriate response. 98) Given matrix A: A= 5 7 -6 -4 9 1 What is the size of A? A) 2 × 3

1 R + R2 → R1 3 1 1 R → R2 2 2

98)

B) 3 × 3

C) 3 × 2

Perform the indicated operations given the matrices. 99) Let A = 1 3 and B = 0 4 ; 3A + B 2 5 -1 6 3 7 A) B) 3 13 5 11 5 21

D) 3

99) C)

3 21 3 33

3 13 1 11

Solve the system of equations by graphing. 100) y = x + 1 y = 3x + 7 10

100)

-10

-5

10 x

-5 -10

A) (-3, -2)

B) (3, 2)

C) (-2, -3)

D) no solution

Perform the operation, if possible. 1 -1 101) Let C = - 3 and D = 3 . Find C - 4D. 2 -2 5 -3 A) -15 B) 9 10 -6 102) Let B = -1 2 6 -3 A) C) 103) A =

101) 5 C) -6 4

, B = 0 -1 4 6

102) B) D)

-3 0 4 -5 2 2 6 -3

Find AB.

103)

B) -4 -9 0 -5

-5 15 -10

. Find -2B.

2 -4 -12 6 -2 4 12 -6 3 -1 5 0

D) 0 20

-9 -4 -5 0

1 0

Find the values of a, b, c, and d that make the matrix equation true. 104) 1 2 a b = 1 2 0 -1 c d 2 4 2 1 A) B) 1 0 C) 5 10 4 2 0 1 -2 -4

-5 0 42 -4

104) D) 0 0 2 5

Solve the system as matrix equations using inverses. 105) A company produces three models of MP3 players, models A, B, and C. Each model A machine requires 3.2 hours of electronics work, 2.8 hours of assembly time, and 4.4 hours of quality assurance time. Each model B machine requires 5.4 hours of electronics work, 2.4 hours of assembly time, and 3.4 hours of quality assurance time. Each model C machine requires 2.2 hours of electronics work, 5.8 hours of assembly time, and 4.8 hours of quality assurance time. There are 303 hours available each week for electronics, 393 hours for assembly, and 416 hours for quality assurance. How many of each model should be produced each week if all available time must be used? A) Model A: 30 B) Model A: 28 C) Model A: 30 D) Model A: 31 Model B: 20 Model B: 22 Model B: 15 Model B: 20 Model C: 45 Model C: 45 Model C: 50 Model C: 44 Solve the problem. 106) Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total income from all 3 investments is $1600. The income from the 5% and 6% investments is the same as the income from the 8% investment. Find the amount invested at each rate. A) $10,000 at 5%, $5000 at 6%, $10,000 at 8% B) $10,000 at 5%, $10,000 at 6%, $5000 at 8% C) $5000 at 5%, $10,000 at 6%, $10,000 at 8% D) $8000 at 5%, $10,000 at 6%, $7000 at 8%

105)

106)

Provide an appropriate response. 107) Solve the linear system corresponding to the following augmented matrix: 3 6 24 2 3 11 A) (0, 0) B) (5, -2) C) (-2, -5) Solve the system of equations by elimination. 108) 2x - 7y = 3 5x - 4y = -6 A) (-2, -1) B) (-2, 1)

107)

D) (-2, 5)

108) C) (2, 1)

D) (2, -1)

Provide an appropriate response. 109) Write the augmented matrix for the system. 8 x1 + 9 x2 = 117

109)

4x1 + 6 x2 = 66

A) 117 9 66 4

8 6

B) 8 4 117 9 6 66

C) 8 9 66 6 4 117

Solve using Gauss-Jordan elimination. 110) 6x + 4y = 0 4x + 2y = -2 A) (-2, 3) B) (3, -2)

D) 8 9 117 4 6 66

110) C) (-2, -3)

Determine whether B is the inverse of A. 2 -3 111) A = 5 3 , B = 3 2 -3 5 A) Yes

D) No solution

111) B) No

Provide an appropriate response. 112) Use the augmented matrix to solve the system: 0.4x1 + 0.9x2 = - 4.9 x1 - 0.3x2 = A) (0, 0)

112)

0.5 B) (-0.1, -0.5)

C) (-1, -5)

Find the inverse, if it exists, of the given matrix. 113) -6 -6 -5 -5 - 5 - 6 11 11 A) - 5 - 6 11 11

5 11

- 6 11

- 5 11

6 11

D) (-5, -1)

113) - 5 11 B)

6 11

5 - 6 11 11

D) Does not exist

Identify the row operation that produces the resulting matrix. 114) 1 -3 4 → 1 -3 4 2 3 1 -1 -6 3 A) R 1 + (-1)R 2 → R 2 B) (-1)R 2 → R 2 C) R 1 + (-1)R 2 → R 1

114)

D) (-1)R 1 + R 2 → R 2

Perform the indicated row operations on the following matrix. 1 -5 4 2 2 5 115) -3R 1 → R 1 A)

1 -5 4 -6 -6 -15

B) -3 15 -12 2 2 5

115)

1 -5 4 -1 17 7

D) -3 -5 4 -6 2 5

Solve the equation for the indicated variable. Assume that the dimensions are such that matrix multiplication and addition are possible and that inverses exist when needed. 116) Solve for Y: XY + ZY = A 116) A) Y = X -1(A - Z) B) Y = A(X + Z) -1 C) Y = (X + Z) -1 A

D) Y = X -1(A - ZY)

Solve using Gauss-Jordan elimination. 117) x1 + x2 = 0 x1 - x 2 = 12 A) (6, -6)

117)

B) (-5, -6)

C) (5, -5)

Find the inverse, if it exists, of the given matrix. 1 1 1 118) 2 1 1 2 2 3 A) -1 -1 -1 -2 -1 -1 -2 -2 -3

D) (-6, -5)

118) B)

1 1 2

1 2

1 3

D) Does not exist -1 1 0 4 -1 -1 -2 0 1

The matrix is the final matrix form for a system of two linear equations in variables x1 and x2. Write the Solution of the system. 119) 1 -4 10 0 0 0 A) No solution

119) B) x1 = 4t + 10 x2 = t for any real number t D) x1 = t - 4

C) x1 = t for any real number t x2 = 10

x2 = t for any real number t

Solve the problem. 120) A Dawn Bakery bakes whole wheat, oat, and rye bread, with mixing, baking, and packaging times, in hours, as shown: Mix 0.04 A = 0.03 0.04

120)

Bake Package 0.07 0.02 Whole wheat 0.05 0.02 Oat 0.06 0.02 Rye

An order is received for 400 loaves of whole wheat bread, 200 loaves of oat bread, and 350 loaves of rye bread. Given that the cost of mixing, baking, and packaging is $14, $25, and$2, respectively, per hour, find matrices B and C so that the product BAC will give the total cost (excluding raw materials) of filling this order. Find the total cost. A) 400 B = 14 25 2 , C = 200 , total cost = $1313 350 B) 14 B = 25 , C = 400 200 350 , total cost = $1313 2 C) B = 400 200 350 , C = 14 25 2 , total cost = $2017 D) 14 B = 400 200 350 , C = 25 , total cost = $2017 2 User row operations to change the matrix to reduced form. 1 0 2 121) -1 1 3 A)

1 0 2 -1 1 3

1 0 2 0 0 5

121) C)

0 1 5 -1 1 3

1 0 2 0 1 5

Write a system of equations associated with the augmented matrix. Do not try to solve. 3 3 5 -2 122) 5 0 7 4 3 6 0 2 A) 3x1 + 3 x2 + 5 x3 = - 2 B) 3x1 + 3 x2 + 5 x3 = 2 5x1 + 7x3 = - 4 3x1 + 6 x2 =-2

5x1 + 7x3 = 4 3x1 + 6 x2 = 2

C) 3x1 - 3 x2 + 5 x3 = - 2

D) 3x1 + 3 x2 + 5 x3 = - 2

5x1 + 7x3 = 4 3x1 + 6 x2 = 2

Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM. 123) There were 35,000 people at a ball game in Atlanta. The day's receipts were $290,000. How many people paid $14 for reserved seats and how many paid $6 for general admission? Let x represent the number of reserved seats and y represent the number of general admission seats. A) 25,000x + 14y = 35,000 B) 20,000x + 14y = 6 x + y = 290,000 x + y = 15,000 C) 15,000x + 14y = 20,000 D) 14x + 6y = 105 x + y = 6 x + y = 35,000 Find the inverse, if it exists, of the given matrix. 0 4 4 124) A = -2 0 8 0 4 0 A) 1 1 0 4 -1 2

-1

1 4

123)

124) B)

0 -

122)

-1

-1 2

-1

1 4

C) Does not exist

-1 2

-1

1 4

0 -

1 4

Identify the row operation that produces the resulting matrix. 2 0 2 125) -2 2 8 → 2 0 2 0 1 5 A)

1 1 R + R → R2 2 1 2 2

125) B) R 1 + R 2 → R 2

1 1 C) - R 1 + - R 2 → R 1 2 2

1 R → R1 2 2

Write the linear system corresponding to the reduced augmented matrix. 1 0 0 4 126) 0 1 1 6 0 0 0 0 A) x1 = -4, x2 = t + 6, x3 = t for any real number t

126)

B) x1 = 4, x2 = t + 6, x3 = 0 C) x1 = 4, x2 = -t + 6, x3 = t for any real number t D) No Solution Solve the problem. 127) Sam and Chad are ticket-sellers at their class play. Sam is selling student tickets for $2.00 each, and Chad selling adult tickets for $5.50 each. If their total income for 24 tickets was $83.00, how many tickets did Sam sell? A) 10 tickets B) 16 tickets C) 14 tickets D) 15 tickets 128) Your screen print operation is doing extremely well at the craft shows. Last week you sold 50 tie -dyed shirts for $15 each, 40 Cheraw-Tech crew shirts for $10 each and 30 handpainted T-shirts for $12 each. Use matrix operations to calculate your total revenue for the week. A) $1510 B) $1750 C) $1151 D) $1480

127)

128)

Answer Key Testname: CHAP 04_14E

1) D 2) B 3) A 4) B 5) -2 -3 4 -9 6) x1 = -1, x2 = -3, x3 = -2 7) THE YELLOW OWL IS HERE 8) W R $160 $230 Store 1 $160 $540 $720 Store 2 9) x1 = -7, x2 = 16 10) x = 1, y = 1.5, z = -2.5; x =-1, y = 1, z = -1 11) Equilibrium price: $45; Equilibrium quantity: 100 12) 295 printers per month 13) no solution 14) 50x1 + 40x2 + 50x3 = 50,000 60x1 + 40x2 + 70x3 = 80,000 x1 - 3x3 = 0 15) (I -M)-1 = 1.6 0.6 0.4 1.4

X = 130 70

16) 1 2 5 0 1 0 -1 -2 -4 17) (-7.765, -3.860) 18) 4x + 6 20 + 3y 8x + 14 40 + 7y 19) x1 = 5 - 4t; x2 = t; x3 = 6 20) (-0.857, 5.286) 21) b 4 - a2b 2 ab 3 - a3 b -ab 3 + a3 b -a2b 2 + a4 22) 4 × 2; a32 = -2; a11 = 5. 23) x1 = 8.85, x2 = -2.78, x3 = -17.31, x4 =11.24 24) x = -3, y = 4; x = -7, y = -5 25) 0.14 0.14 0.15 M = 0.18 0.15 0.17 0.17 0.13 0.11 26) $3,010; 20.87% 25

Answer Key Testname: CHAP 04_14E

27) 5 ounces rice, 7 ounces broccoli, 8 ounces fish 28) Park 1: 100 experienced workers Park 2: 40 experienced workers Park 3: 80 experienced workers 29) 2 model A, 8 model B, and 0 model C trucks; or 4 model A, 5 model B, and 1 model C trucks; or 6 model A, 2 model B, and 2 model C trucks 30) $44 billion agriculture, $52 billion manufacturing 31) 50x + 50y + 75z = 12,000 75x + 60y + 100z = 14,500 35x + 30y + 45z = 6,000 32) 0.6x + 0.5y + 0.3z = 11 0.3x + 0.3y + 0.4z = 15 0.1x + 0.2y + 0.3z = 14 33) x1 + x2 = 9,000 0.13x 1 + 0.08x 2 =

810

34) x1 = -7; x2 = 6; x3 = 1 35) 4 -1 -3 1 36) x1 = 35; x2 = 40 37) C 38) B 39) C 40) B 41) D 42) A 43) D 44) C 45) A 46) B 47) A 48) B 49) C 50) C 51) D 52) B 53) D 54) D 55) B 56) D 57) B 58) C 59) B 60) D 61) C 62) C 26

Answer Key Testname: CHAP 04_14E

63) A 64) B 65) A 66) D 67) B 68) B 69) C 70) B 71) C 72) D 73) A 74) B 75) A 76) B 77) D 78) D 79) D 80) B 81) A 82) B 83) D 84) A 85) C 86) A 87) A 88) C 89) B 90) B 91) D 92) C 93) B 94) D 95) C 96) C 97) C 98) A 99) B 100) A 101) A 102) A 103) A 104) C 105) A 106) A 107) D 108) A 109) D 110) A 111) A 27

Answer Key Testname: CHAP 04_14E

112) C 113) D 114) A 115) B 116) C 117) A 118) C 119) B 120) D 121) D 122) D 123) D 124) D 125) A 126) C 127) C 128) A

CHAPTER 5

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) A vineyard produces two special wines a white, and a red. A bottle of the white wine requires 14 pounds of grapes and 1 hour of processing time. A bottle of red wine requires 25 pounds of grapes and 2 hours of processing time. The vineyard has on hand 2,198 pounds of grapes and can allot 160 hours of processing time to the production of these wines. A bottle of the white wine sells for $11.00, while a bottle of the red wine sells for $20.00. How many bottles of each type should the vineyard produce in order to maximize gross sales? (Solve using the geometric method.) Provide an appropriate response. 2) Using a graphing calculator as needed, maximize P = 310x1 + 470x2 subject to 250x1 + 450x2 301x1 + 390x2

≤ 4190 ≤ 3700

382x1 + 289x2 ≤ 4404 x1, x2 ≥ 0 Give the answer to two decimal places. 3) The corner points for the bounded feasible region determined by the system of inequalities:

2x1 + 5x2 ≤ 20 x1 + x2 ≤ 7 x1, x2 ≥ 0 are O = (0, 0), A = (0, 4), B = (5, 2) and C = (7, 0). Find the optimal solution for the objective profit function: P(x) = 3x1 + 7x2 Solve the problem. 4) Formulate the following problem as a linear programming problem (DO NOT SOLVE):A dietitian can purchase an ounce of chicken for $0.25 and an ounce of potatoes for $0.02. Each ounce of chicken contains 13 units of protein and 24 units of carbohydrates. Each ounce of potatoes contains 5 units of protein and 35 units of carbohydrates. The minimum daily requirements for the patients under the dietitian's care are 45 units of protein and 58 units of carbohydrates. How many ounces of each type of food should the dietitian purchase for each patient so as to minimize costs and at the same time insure the minimum daily requirements of protein and carbohydrates? (Let x1 equal the number of ounces of chicken and x2 the number of ounces of potatoes purchased per patient.)

Provide an appropriate response. 5) Find the coordinates of the corner points of the solution region for: 3x + 2y ≥ 54 4x + 5y ≤ 100 x ≥0 y ≥0 Solve the problem. 6) The Southern States Ring Company designs and sells two types of rings: the brass and the aluminum. They can produce up to 24 rings each day using up to 60 total man-hours of labor per day. It takes 3 man-hours to make one brass ring and 2 man-hours to make one aluminum ring. How many of each type of ring should be made daily to maximize the company's profit, if the profit on a brass ring is $40 and on an aluminum ring is $35?

Give the mathematical formulation of the linear programming problem. Graph the feasible region described by the constraints and find the corner points. Do not attempt to solve. 7) Suppose an horse feed to be mixed from soybean meal and oats must contain at least 7) 200 lb of protein and 40 lb of fat. Each sack of soybean meal costs $20 and contains 60 lb of protein and 10 lb of fat. Each sack of oats costs $10 and contains 20 lb of protein and 5 lb of fat. How many sacks of each should be used to satisfy the minimum requirements at minimum cost? Solve the problem. 8) Formulate the following problem as a linear programming problem (DO NOT SOLVE).A company which produces three kinds of spaghetti sauce has two plants. The East plant produces 3,500 jars of plain sauce, 6,500 jars of sauce with mushrooms, and 3,000 jars of hot spicy sauce per day. The West plant produces 2,500 jars of plain sauce, 2,000 jars of sauce with mushrooms, and 1,500 jars of hot spicy sauce per day. The cost to operate the East plant is $8,500 per day and the cost to operate the West plant is $9,500 per day. How many days should each plant operate to minimize cost and to fill an order for at least 8,000 jars of plain sauce, 9,000 jars of sauce with mushrooms, and 6,000 jars of hot spicy sauce? (Let x1 equal the number of days East plant should operate and x2 the

number of days West plant should operate.) 9) Formulate the following problem as a linear programming problem (DO NOT SOLVE):A steel company produces two types of machine dies, part A and part B. Part A requires 6 hours of casting time and 4 hours of firing time. Part B requires 8 hours of casting time and 3 hours of firing time. The maximum number of hours per week available for casting and firing are 85 and 70, respectively. The company makes a $2.00 profit on each part A that it produces, and a $6.00 profit on each part B that it produces. How many of each type should the company produce each week in order to maximize its profit? (Let x1 equal the number of A parts and x2 equal the number of B parts produced each week.)

Provide an appropriate response. 10) Refer to the following system of linear inequalities associated with a linear programming problem: Maximize

10)

P = 3x1 + 7x2

subject to 5x1 + x2 ≤ 28 2x1 + x2 ≤ 13 x1 + x2 ≤ 0 x1, x2 ≥ 0 (A) Determine the number of slack variable that must be introduced to form a system of problem constraint equations. (B) Determine the number of basic variables associated with this system. 11) Using a graphing calculator as needed, maximize P = 524x1 + 479x2 subject to 265x1 + 320x2 350x1 + 345x2

11)

≤ 3,390 ≤ 3,795

400x1 + 316x2 ≤ 4,140 x1, x2 ≥ 0 Give the answer to two decimal places. 12) The corner points for the bounded feasible region determined by the system of inequalities: 5x1 + 2x2 ≤ 40 x1 + 3x2 ≤ 21 x1, x2 ≥ 0 are O = (0, 0), A = (0, 7), B = (6, 5) and C = (8, 0). Find the optimal solution for the objective profit function: P = 5x1 + 5x2

12)

Solve the problem. 13) Formulate the following problem as a linear programming problem (DO NOT SOLVE):A small accounting firm prepares tax returns for two types of customers: individuals and small businesses. Data is collected during an interview. A computer system is used to produce the tax return. It takes 2.5 hours to enter data into the computer for an individual tax return and 3 hours to enter data for a small business tax return. There is a maximum of 40 hours per week for data entry. It takes 20 minutes for the computer to process an individual tax return and 30 minutes to process a small business tax return. The computer is available for a maximum of 900 minutes per week. The accounting firm makes a profit of $125 on each individual tax return processed and a profit of $210 on each small business tax return processed. How many of each type of tax return should the firm schedule each week in order to maximize its profit? (Let x1

13)

equal the number of individual tax returns and x2 the number of small business tax returns.) Provide an appropriate response. 14) Solve the following linear programming problem by determining the feasible region on the graph below and testing the corner points: Minimize

14)

C = x1 + 6x2

subject to 3x1 + 4x2 ≥ 36 2x1 + x2 ≤ 14 x1, x2 ≥ 0 y

10 (4, 6)

x1 is shown on the x-axis and x2 on the y-axis. Give the mathematical formulation of the linear programming problem. Graph the feasible region described by the constraints and find the corner points. Do not attempt to solve. 15) A math camp wants to hire counselors and aides to fill its staffing needs at minimum 15) cost. The monthly salary of a counselor is $2400 and the monthly salary of an aide is $1100. The camp can accommodate up to 45 staff members and needs at least 30 to run properly. They must have at least 10 aides, and at most twice as many aides as counselors. How many counselors and how many aides should the camp hire to minimize cost?

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or unbounded. 16) y ≤ -2x - 2 16) y ≥ x +9 y

10 5

-10

-5

10 x

5 -5 -10

A) Unbounded

B) Unbounded 10

-10

-5

10 x

-10

-5

-10

C) Unbounded

10 x

D) Unbounded 10

-10

-5

10 x

-10

-5

-10

Graph the inequality. 17) y < - 2

17) y

10 5

-10

-5

10 x

5 -5 -10

B) 10

-10

-5

10 x

-10

-5

-10

10 x

D) 10

-10

-5

10 x

-10

-5

-10

Solve the problem. 18) The Old-World Class Ring Company designs and sells two types of rings: the BRASS and the GOLD. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one BRASS ring and 2 man-hours to make one GOLD ring. How many of each type of ring should be made daily to maximize the company's profit, if the profit on a BRASS ring is $40 and on an GOLD ring is $30? A) 14 BRASS and 14 GOLD B) 14 BRASS and 10 GOLD C) 12 BRASS and 12 GOLD D) 10 BRASS and 14 GOLD

18)

Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or unbounded. 6

19)

x ≥ 2y x + 2y ≤ 1

19) y

10 5

-10

-5

10 x

5 -5 -10

A) Unbounded

B) Bounded 10

-10

-5

10 x

-10

-5

-10

C) Unbounded

10 x

D) Unbounded 10

-10

-5

10 x

-10

-5

-10

Use graphical methods to solve the linear programming problem. 20) Minimize z = 4x + 5y subject to: 2x - 4y ≤ 10 2x + y ≥ 15 x≥0 y≥0 10

20)

10 x

-10

A) Minimum of 39 when x = 1 and y = 7 C) Minimum of 33 when x = 7 and y = 1

B) Minimum of 75 when x = 0 and y = 15 D) Minimum of 20 when x = 5 and y = 0

21) Suppose an horse feed to be mixed from soybean meal and oats must contain at least 100 lb of protein, 20 lb of fat, and 9 lb of mineral ash. Each 100-lb sack of soybean meal costs $20 and contains 50 lb of protein, 10 lb of fat, and 8 lb of mineral ash. Each 100 -lb sack of oats costs $10 and contains 20 lb of protein, 5 lb of fat, and 1 lb of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum cost? A)

21)

7 sacks of soybeans and 5 sacks of oats 3 6

B) 1 3 sacks of soybeans and 1 9 sacks of oats 11 11 C) 0 sacks of soybeans and 2 sacks of oats D) 2 sacks of soybeans and 0 sacks of oats Define the variable(s) and translate the sentence into an inequality. 22) Enrollment is below 8000 students. A) Let e = student enrollment; e ≤ 8000 B) Let e = student enrollment; e ≥ 8000 C) Let e = student enrollment; e < 8000 D) Let e = student enrollment; e > 8000 Solve the problem. 23) A salesperson has two job offers. Company A offers a weekly salary of $180 plus commission of 6% of sales. Company B offers a weekly salary of $360 plus commission of 3% of sales. What is the amount of sales above which Company A's offer is the better of the two? A) $3000 B) $6100 C) $12,000 D) $6000

22)

23)

Graph the constant-profit lines through (3, 2) and (5, 3). Use a straightedge to identify the corner point(s) where the maximum profit occurs for the given objective function. y 14 13 (0, 12) 12 11 10 9 8 7 6 5 4 3 (5, 3) 2 (3, 2) 1

(8, 4)

(9, 0)

1 2 3 4 5 6 7 8 9 1011121314 x

24) P = x + y A) Max P = 8 at x = 5 and y = 3 B) Max P = 9 at x = 9 and y = 0, at x = 8 and y = 4, and at every point on the line segment joining the preceding two points. C) Max P = 12 at x = 0 and y = 12, at x = 8 and y = 4, and at every point on the line segment joining the preceding two points. D) Max P = 5 at x = 3 and y = 2 Use graphical methods to solve the linear programming problem. 25) Minimize z = 2x + 4y subject to: x + 2y ≥ 10 3x + y ≥ 10 x≥0 y≥0 10

24)

25)

10 x

-10

A) Minimum of 0 when x = 0 and y = 0 B) Minimum of 20 when x = 10 and y = 0 C) Minimum of 20 when x = 2 and y = 4, as well as when x = 10 and y = 0, and all points in between D) Minimum of 20 when x = 2 and y = 4 Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or unbounded. 9

26) y > 4 x≥5

26) y

10 5

-10

-5

10 x

5 -5 -10

A) Bounded

B) Bounded 10

-10

-5

10 x

-10

-5

-10

C) Unbounded

10 x

D) Unbounded 10

-10

-5

10 x

-10

-5

-10

27) x ≤ -4 y>2 3x + y < 6

27)

25 x

-25

A) Bounded

B) Bounded 25

25 x

-25

25 x

-25

C) Unbounded

D) Unbounded 25

25 x

-25

25 x

-25

Define the variable(s) and translate the sentence into an inequality. 28) Sales of wheat bread are at least $2000 greater than sales of white bread. A) Let e = wheat bread sales; let i = white bread sales; w + i ≥ $2000 B) Let e = wheat bread sales; let i = white bread sales; w ≤ i + $2000 C) Let e = wheat bread sales; let i = white bread sales; w ≥ i + $2000 D) Let e = wheat bread sales; let i = white bread sales; i ≥ w + $2000

28)

(8, 4)

(9, 0)

1 2 3 4 5 6 7 8 9 1011121314 x

29) P = 5x + y A) Max P = 45 at x = 9 and y = 0 B) Max P = 44 at x = 8 and y = 4 C) Max P = 44 at x = 8 and y = 4, at x = 0 and y = 12, and at every point on the line segment joining the preceding two points. D) Max P = 45 at x = 9 and y = 0, at x = 8 and y = 4, and at every point on the line segment joining the preceding two points.

29)

Graph the inequality. 30) 4x - 2y ≤ 8

30) y

10 5

-10

-5

10 x

5 -5 -10

B) 10

-10

-5

10 x

-10

-5

-10

10 x

D) 10

-10

-5

10 x

-10

-5

-10

Solve the problem. 31) Jim has gotten scores of 78 and 69 on his first two tests. What score must he get on his third test to keep an average of 70 or greater? A) At least 73.5 B) At least 72.3 C) At least 62 D) At least 63

31)

Graph the inequality. 32) 2x + y ≤ 1

32) y

10 5

-10

-5

10 x

5 -5 -10

B) 10

-10

-5

10 x

-10

-5

-10

10 x

D) 10

-10

-5

10 x

-10

-5

-10

Use graphical methods to solve the linear programming problem. 33) Maximize z = 6x + 7y subject to: 2x + 3y ≤ 12 2x + y ≤ 8 x≥0 y≥0 10

33)

10 x

-10

A) Maximum of 52 when x = 4 and y = 4 C) Maximum of 32 when x = 3 and y = 2

B) Maximum of 24 when x = 4 and y = 0 D) Maximum of 32 when x = 2 and y = 3

Solve the problem. 34) Company A rents copiers for a monthly charge of $300 plus 10 cents per copy. Company B rents copiers for a monthly charge of $600 plus 5 cents per copy. What is the number of copies above which Company A's charges are the higher of the two? A) 12,000 copies B) 6100 copies C) 6000 copies D) 3000 copies Use graphical methods to solve the linear programming problem. 35) A math camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average monthly salary of a counselor is $2400 and the average monthly salary of an aide is $1100. The camp can accommodate up to 45 staff members and needs at least 30 to run properly. They must have at least 10 aides, and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the camp hire to minimize cost? A) 18 counselors and 12 aides C) 12 counselors and 18 aides

34)

35)

B) 35 counselors and 10 aides D) 27 counselors and 18 aides

Graph the solution set of the system of linear inequalities and indicate whether the solution region is bounded or unbounded.

36) y ≤ -2x + 2 y ≥ x-9

36) y

10 5

-10

-5

10 x

5 -5 -10

A) Bounded

B) Unbounded 10

-10

-5

10 x

-10

-5

-10

C) Bounded

10 x

D) Unbounded 10

-10

-5

10 x

-10

-5

-10

Graph the inequality. 37) x + 5 y ≥ 2

37) y

10 5

-10

-5

10 x

5 -5 -10

B) 10

-10

-5

10 x

-10

-5

-10

10 x

D) 10

-10

-5

10 x

-10

-5

-10

Use graphical methods to solve the linear programming problem. 38) Maximize z = 8x + 12y subject to: 40x + 80y ≤ 560 6x + 8y ≤ 72 x≥0 y≥0 10

10 x

-10

A) Maximum of 96 when x = 9 and y = 2 C) Maximum of 92 when x = 4 and y = 5

B) Maximum of 120 when x = 3 and y = 8 D) Maximum of 100 when x = 8 and y = 3

38)

Answer Key Testname: CHAP 05_14E

1) 132 bottles of white wine, and 14 bottles of red wine 2) P = 4415.3 at x1 = 0.81, x2 = 8.86. 3) Maximum occurs at (5, 2) and is 29. 4) Minimize C = 0.25x 1 + 0.02x 2 subject to 13x1 + 5x2 ≥ 45 24x1 + 35x2 ≥ 58 x1, x2 ≥ 0 5) (18, 0), (25, 0), (10, 12) 6) Maximize P = 40x + 35y Subject to x + y ≤ 24 3x + 2y ≤ 60 x ≥ 0, y ≥ 0 y 45 40 35 30 25

20 15

10 5 D A

10 15 20 25 30 35 40 45 x

Corner Points: A = (0, 0) B = (0, 24) C = (12, 12) D = (20, 0)

Answer Key Testname: CHAP 05_14E

7) Minimize C = 20x + 10y Subject to 60x + 20y ≥ 200 10x + 5y ≥ 40 x ≥ 0, y ≥ 0 y 10

9 8 7 6 5 4

3 2 1 -1

C 1

9 10 x

Corner Points:

8) Minimize

A = (0, 10) B = (2, 4) C = (4, 0) C = 8,500x 1 + 9,500x 2

subject to 3,500x 1 + 2,500x 2 ≥ 8,000 6,500x 1 + 2,000x 2 ≥ 9,000 3,000x 1 + 1,500x 2 ≥ 6,000 x1, x2 ≥ 0 9) Maximize

P = 2x1 + 6x2

subject to 6x1 + 8x2 ≤ 85 4x1 + 3x2 ≤ 70 x1, x2 ≥ 0 10) (A) 3; (B) 3 11) P = 5587.72 at x1 = 8.36, x2 = 2.52. 12) Maximum occurs at (6, 5) and is 55. 13) Maximize P = 125x1 + 210x2 subject to 2.5x 1 + 3x2 ≤ 40 20x1 + 30x2 ≤ 900 x1, x2 ≥ 0 14) Minimum at (4, 6).

Answer Key Testname: CHAP 05_14E

15) Minimize C = 2400x + 1100y Subject to 30 ≤ x + y ≤ 45 y ≥ 10 y ≤ 2x x≥0 y 45 40 35

30 25 20

15 C

5 5

10 15 20 25 30 35 40 45 x

Corner Points: A = (10, 20) B = (15, 30) C = (35, 10) D = (20, 10) 16) A 17) C 18) C 19) D 20) C 21) D 22) C 23) D 24) C 25) C 26) C 27) D 28) C 29) A 30) D 31) D 32) A 33) C 34) C 35) C 36) B 37) C 38) D

CHAPTER 6

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Consider the linear programming problem: Maximize

P = 3x1 + 7x2

subject to x1 + x2 ≥ 6 2x1 + x2 ≥ 5 x1 + 3x2 ≤ 15 x1, x2 ≥ 0 (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem. (DO NOT SOLVE.) Solve the problem. 2) A stereo manufacturer makes three types of stereo systems, I, II, and III, with profits of $20, $30, and $40, respectively. No more than 100 type -I systems can be made per day. Type -I systems require 5 man-hours, and the corresponding numbers of man-hours for types II and III are 10 and 15, respectively. If the manufacturer has available 2000 man-hours per day, determine the number of units from each system that must be manufactured in order to maximize profit. Compute the corresponding profit. 3) Consider the following linear programming problem. x - 2z ≤ 10 Maximize P = 3x + 2y + 5z subject to x + y + 6z ≤ 80 x ≥ 0, y ≥ 0, z ≥ 0. The associated final simplex tableau is as follows: x y z u v P 1 0 -2 1 0 0 10 0 1 8 -1 1 0 70 0 0 5 1 2 1 170

Find the optimal solution of the minimization problem . Provide an appropriate response. 4) For the following initial simplex tableau, identify the basic and nonbasic variables. Find the pivot element, the entering and exiting variables, and perform one pivot operation. x

4 1 -15 -10

0 0

1 0

0 1

4 0

5) State whether the optimal solution has been found, an additional pivot is required, or there is no solution for the problem corresponding to the following simplex tableau:

x1 x2 x3 s 1 s 2 s 3 P 1 0 0 3 1 1 0 8 0 1 0 -2 0 0 0 40 -2 0 1 2 1 0 0 5 1 0 0 4 2 0 1 54 Solve the problem. 6) Big Round Cheese Company has on hand 45 pounds of Cheddar and 49 pounds of Brie each day. It prepares two Christmas packagesthe "Holiday" box, which has 5 pounds of Cheddar and 2 pounds of Brie, and the "Noel" box, which contains 2 pounds of Cheddar and 7 pounds of Brie. Profit on each Holiday assortment is $6, profit on each Noel assortment is $8.

(a) Give the optimal production schedule and the resulting maximum profit. (b) How much excess Cheddar and excess Brie remain each day if this plan is followed? 7) Consider the following linear programming problem. 2x + 3y ≤ 9 Maximize P = 5x + 6y subject to x + y≤4 x ≥ 0, y ≥ 0 The associated final simplex tableau is as follows: x y u v P 0 1 1 -2 0 1 1 0 -1 3 0 3 0 0 1 3 1 21

Find the optimal solution of the minimization problem. Provide an appropriate response. 8) State whether the optimal solution has been found, an additional pivot is required, or there is no solution for the modified problem corresponding to the following simplex tableau: x1 x2 x3 s 1 a1 s 2 a2 a3 P 0 1 0 0 0 2 1 1 0 5 1 0 -1 0 1 -1 0 1 0 2 0 0 -3 1 2 1 -1 0 0 8 0 0 -M + 3 0 0 4 M - 5 M + 2 1 20

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 9) Write the basic solution for the following simplex tableau:

x1 x2 s 1 s 2 s 3 P 1 2 1 0 0 0 10 2 3 0 -

1 1 0 2

-2 0 -1 0 1 0 4 14 0 6 0 0 1 120 A) x1 = 0, x2 = 10, s 1 = 0, s 2 = 40, s 3 = 4, P = 120 B) x1 = 0, x2 = 10, s 1 = 1, s 2 = 40, s 3 = 4, P = 120 C) x1 = 0, x2 = 10, s 1 = 0, s 2 = 40, s 3 = 4, P = -120 D) x1 = 5, x2 = 10, s 1 = 0, s 2 = 40, s 3 = 4, P = 120 10) State the dual problem. Minimize C = 6 x1 + 3 x2

10)

subject to: 3x1 + 2x2 ≥ 34 2x1 + 5 x2 ≥ 43 x1 , x2 ≥ 0 A) Maximize P = 34y 1 +43y 2

B) Maximize P = 43 y 1 +34y 2

subject to: 3y 1 + 2y 2 ≥ 6 2y 1 + 5 y 2≥ 3

subject to: 2y 1 + 3 y 2 ≥ 6 5y 1 + 2y 2 ≥ 3

y 1, y 2 ≥ 0

C) Maximize P = 43 y 1 +34y 2

D) Maximize P =34y 1 + 43 y 2

subject to: 2y 1 + 3 y 2 ≤ 6 5y 1 + 2y 2 ≤ 3

subject to: 3y 1 + 2y 2 ≤ 6 2y 1 + 5 y 2 ≤ 3

y 1, y 2 ≥ 0

Solve the problem. 11) A linear programming problem has 4 decision variables, x1 to x4, and 8 problem constraints. How many rows are there in the table of basic solutions of the associated e -system? A) 11,880 B) 248 C) 495 D) 24

11)

Provide an appropriate response. 12) Formulate the dual problem for the linear programming problem: Minimize

12)

C = 22x1 + 15x2

subject to x1 + 2x2 ≥ 3 x1 + x2 ≥ 2 x1, x2 ≥ 0 A) Maximize P = 3y 1 + 2y 2

B) Maximize

subject to

C) Maximize

P = 22y 1 + 15y 2

subject to y 1 + y 2 ≥ 22

y 1 + y 2 ≤ 22

2y 1 + y 2 ≥ 15

2y 1 + y 2 ≤ 15

y 1, y 2 ≥ 0

P = 3y 1 + 2y 2

D) Maximize

subject to

P = 3y 1 + 2y 2

subject to y 1 + y 2 ≤ 22

y1 + y2 ≥ 3

2y 1 + y 2 ≤ 15

2y 1 + y 2 ≥ 2

y 1, y 2 ≥ 0

13) Formulate the dual problem for the linear programming problem:

13)

Minimize C = 3x1 + x2 subject to 2x1 + 3x2 ≥ 60 x1 + 4x2 ≥ 40 x1, x2 ≥ 0 A) Maximize

P = 3y 1 + y 2

B) Maximize

subject to

C) Maximize

P = 3y 1 + y 2

subject to 2y 1 + y 2 ≤ 3

2y 1 + y 2 ≤ 3

3y 1 + 4y 2 ≤ 1

3y 1 + 4y 2 ≥ 1

y 1, y 2 ≥ 0

P = 60y 1 + 40y 2

D) Maximize

subject to

P = 60y 1 + 40y 2

subject to 2y 1 + y 2 ≥ 3

2y 1 + y 2 ≤ 3

3y 1 + 4y 2 ≥ 1

3y 1 + 4y 2 ≤ 1

y 1, y 2 ≥ 0

14) Formulate the dual problem for the linear programming problem Minimize

14)

C = 4x1 + 7x2

subject to x1 + x2 ≥ 5 x1 + 2x2 ≤ 18 3x1 + x2 ≥ 8 x1, x2 ≥ 0 A) Maximize

P = 5y 1 + 18y 2 - 8y 3

B) Maximize

subject to

C) Maximize

P = 5y 1 - 18y 2 + 8y 3

subject to y 1- 2y 2 + y 3 ≤ 7

y 1- 2y 2 + y 3 ≤ 7

y 1- y 2 + 3y 3 ≤ 4

y 1, y 2, y 3 ≥ 0

P = 5y 1 - 18y 2 + 8y 3

D) Maximize

subject to

P = 5y 1 - 18y 2 + 8y 3

subject to y 1- 2y 2 + y 3 ≤ 7

y 1- 2y 2 + y 3 ≥ 7

y 1- y 2 + 3y 3 ≤ 4

y 1- y 2 + 3y 3 ≥ 4

y 1, y 2, y 3 ≤ 0

y 1, y 2, y 3 ≥ 0

Convert the given i-system to an e-system using slack variables. Then construct a table of all basic solutions of the e-system. For each basic solution, indicated whether or not it is feasible. 15) 3x1 + 8x2 ≤ 24

15)

x1, x2 ≥ 0 A) 3x1 + 8x2 + s 1 = 24

B) 3x1 + 8x2 + s 1 = 24

x1 x2 s 1 (A) 0 0 24 (B) 0 3 0 (C) 8 0 0

x1 x2 s 1 (A) 0 0 24 (B) 0 8 0 (C) 3 0 0

Feasible: (A), (B), (C) Not feasible: none C) 3x1 + 8x2 + s 1 ≤ 24

Feasible: (A), (B), (C) Not feasible: none D) 3x1 + 8x2 + s 1 = 24

x1 x2 s 1 (A) 0 0 24 (B) 0 3 0 (C) 8 0 0

x1 x2 s 1 (A) 0 0 24 (B) 0 -3 0 (C) -8 0 0

Feasible: (B), (C) Not feasible: (A)

Feasible: (A) Not feasible: (B), (C)

Solve the problem.

16) Formulate the following problem as a linear programming problem (DO NOT SOLVE):A shoe company is introducing a new line of running shoes. The marketing division decides to promote the line in a particular city. The promotion will consist of newspaper, radio, and television ads. Each newspaper ad will cost $120, each television ad will cost $370, and each radio ad will cost $210. The company wants to spend at most half their money on newspaper ads. The marketing division believes that each newspaper ad will reach 4,700 men and 3,700 women, each television ad will reach 7,600 men and 6,600 women, and each radio ad will reach 4,500 men and 5,500 women. The promotion will be considered successful if it reaches at least 400,000 men and 200,000 women. How should the company divide its money between the newspaper, television, and radio ads so as to insure a successful promotion at a minimum cost? (Let x1 equal the number of newspaper ads, x2 equal the number of television ads, and x3 equal the number of radio ads purchased in the promotion.) A) Minimize C = -120x1 - 370x2 + 210x3 subject to 4,700x 1 + 7,600x 2 + 4,500x 3 ≤ 400,000 3,700x 1 + 6,600x 2 + 5,500x 3 ≤ 200,000 120x1 - 370x2 - 210x3 ≤ 0 x1, x2, x3 ≥ 0 B) Minimize

C = -120x1 - 370x2 + 210x3

subject to 4,700x 1 + 7,600x 2 + 4,500x 3 ≥ 400,000 3,700x 1 + 6,600x 2 + 5,500x 3 ≥ 200,000 120x1 - 370x2 - 210x3 ≤ 0 x1, x2, x3 ≥ 0 C) Maximize C = 120x1 + 370x2 + 210x3 subject to 4,700x 1 + 7,600x 2 + 4,500x 3 ≥ 400,000 3,700x 1 + 6,600x 2 + 5,500x 3 ≥ 200,000 120x1 - 370x2 - 210x3 ≤ 0 x1, x2, x3 ≥ 0 D) Minimize

C = 120x1 + 370x2 + 210x3

subject to 4,700x 1 + 7,600x 2 + 4,500x 3 ≥ 400,000 3,700x 1 + 6,600x 2 + 5,500x 3 ≥ 200,000 120x1 - 370x2 - 210x3 ≤ 0 x1, x2, x3 ≥ 0

16)

Provide an appropriate response. 17) Formulate the dual problem for the linear programming problem: Minimize

17)

C = 3x1 + x2

subject to 2x1 + 3x2 ≥ 60 x1 + 4x2 ≥ 40 x1, x2 ≥ 0 A) Maximize P = 60y 1 + 40y 2

B) Maximize

subject to

P = 3y 1 + y 2

subject to 2y 1 + y 2 ≥ 3

2y 1 + y 2 ≤ 3

3y 1 + 4y 2 ≥ 1

3y 1 + 4y 2 ≤ 1

y 1, y 2 ≥ 0

C) Maximize

P = 3y 1 + y 2

D) Maximize

subject to

P = 60y 1 + 40y 2

subject to 2y 1 + y 2 ≥ 3

2y 1 + y 2 ≤ 3

3y 1 + 4y 2 ≥ 1

3y 1 + 4y 2 ≤ 1

y 1, y 2 ≥ 0

Refer to the table of the six basic solutions to the e-system. 18) 4x1 + 5x2 + s 1 = 60 x1 + 2x2

18)

+ s 2 = 18

x1 x2 s 1 s 2 (A) 0 0 60 18 (B) 0 12 0 -6 (C) 0 9 15 0 (D) 15 0 0 3 (E) 18 0 -12 0 (F) 10 4 0 0 In basic solution (B), identify the basic and nonbasic variables and determine if the solution is feasible or not feasible. A) Basic variables: x2, s 2 B) Basic variables: x2, s 2 Nonbasic variables: x1, s 1

Nonbasic variables: x1, s 1

Solution is feasible. C) Basic variables: x2, s 1

Solution is not feasible. D) Basic variables: x1, s 2

Nonbasic variables: x1, s 2

Nonbasic variables: x2, s 1

Solution is not feasible.

Provide an appropriate response. 19) Write the simplex tableau, label the columns and rows for the linear programming problem: Maximize P = 4x1 + x2 subject to

19)

2x1 + 5 x2 ≤ 9 3x1 + 3 x2 ≤ 2 x1, x2 ≥ 0

x1 x2 s 1 s 2 2 5 1 0 3 3 0 1 -4 -1 0 0

0 0 1

B) 2 9 0 D) 9 2 0

x1 x2 s 1 s 2 2 5 1 0 3 3 0 1 4 1 0 0

0 0 1

9 2 0

2 9 0

20) Formulate the dual problem for the linear programming problem: Minimize

20)

C = 6x1 + x2 + 5x3

subject to 8x1 + x2 ≥ 2 8x2 + 5x3 ≥ 16 x1, x2, x3 ≥ 0 A) Maximize P = 2y 1 + 16y 2

B) Maximize

subject to

C) Maximize

P = 16y 1 + 2y 2

subject to 8y 1 ≥ 6

8y 1 ≤ 6

y 1 + 8y 2 ≥ 1

y 1 + 8y 2 ≤ 1

5y 2 ≤ 5

y 1, y 2 ≥ 0

P = 2y 1 + 16y 2

D) Maximize

subject to

P = 2y 1 + 16y 2

subject to 8y 1 ≤ 6

8y 1 ≤ 6

y 1 + 8y 2 ≤ 1

5y 2 ≤ 5

y 1, y 2 ≥ 0

y 1, y 2 ≤ 0

21) Use the simplex method to solve the linear programming problem. Minimize w = 5 y 1 + 2y 2 subject to:

21)

y 1 + y 2 ≥ 19.5 2y 1 + y 2 ≥ 24 y 1 ≥ 0, y 2 ≥ 0

A) 30.5 when y 1 = 4.5 and y 2 = 0

B) 31.5 when y 1 = 1 and y 2 = 2

C) 54 when y 1 = 24 and y 2 = 1

D) 48 when y 1 = 0 and y 2 = 24

22) Convert the inequality to a linear equation by adding a slack variable. x1 + 6 x2 ≤ 35 A) x1 + 6 x2 + s 1 + 35 = 0

B) x1 + 6 x2 + s 1 < 35

C) x1 + 6 x2 + s 1 = 35

D) x1 + 6 x2 + s 1 ≤ 35

23) Use the big M method to solve the following linear programming problem.

22)

23)

Maximize P = 6x1 + 4x2 + x3 subject to x1 + 3x2 + 6x3 ≤ 18 x1 + x2 + x3 ≥ 5 x1, x2, x3 ≥ 0 A) Max P = 108 at x1 = 18, x2 = 0, x3 = 9 B) Max P = 108 at x1 = 0, x2 = 0, x3 = 18 C) Max P = 108 at x1 = 0, x2 = 18, x3 = 0 D) Max P = -108 at x1 = 18, x2 = 0, x3 = 0 E) Max P = 108 at x1 = 18, x2 = 0, x3 = 0 Pivot once about the circled element in the simplex tableau, and read the solution from the result. 24)

A) x1 = 24, s 2 = -16, z = 24; x2, x3 , s 1 = 0

B) x1 = 24, s 2 = -16, z = -24; x2, x3 , s 2 = 0

C) x1 = 48, s 2 = -16, z = -48; x2, x3 , s 1 = 0

D) x1 = 48, s 2 = 16, z = 48; x2, x3 , s 1 = 0

24)

Write the e-system obtained via slack variables for the linear programming problem. 25) Maximize P = 3 x1 + 5 x2

25)

subject to: 9x1 + 2x2 ≤ 30 x1 + 8 x2 ≤ 40 with:

x1 ≥ 0, x2 ≥ 0 A) 9x1 + 2x2 + s 1 = 30

B) 9x1 + 2x2 = s 1 + 30

x1 + 8 x2 + s 2 = 40 C) 9x1 + 2x2 + s 1 ≤ 30

x1 + 8 x2 = s 2 + 40 D) 9x1 + 2x2 + s 1 = 30

x1 + 8 x2 + s 2 ≤ 40

x1 + 8 x2 + s 1 = 40

Provide an appropriate response. 26) Use the big M method to find the optimal solution to the problem. Maximize P = 6 x1 + 2x2 subject to

26)

x1 + 2x2 ≤ 20 2x1 + x2 ≤ 16 x1 + x2 ≥ 9 x1, x2 ≥ 0

A) max P = 46 at x1 = 2 , x2 = 7

B) max P = 46 at x1 = 7 , x2 = 2

C) max P = 46 at x1 , x2 = 2

D) max P = - 46 at x1 = 2 , x2 = 7

Refer to the table of the six basic solutions to the e-system. 27) The following chart depicts basic solutions to a system of linear inequalities. Which of the solutions is NOT feasible? x1 x2 s 1 s 2 (A) 0 0 11 6 11 4 (B) 0 0 3 3 (C) 0 3 11 (D) 0 2

2 0

0 1 2

(E) 6 0 -1 0 (F) 4 1 0 0 A) (B) C) (B) and (E)

B) (A), (D), (E), (F) D) (A), (B), (C)

27)

Provide an appropriate response. 28) Solve the following linear programming problem by applying the simplex method to the dual problem: Minimize

28)

C = 3x1 + 2x2

subject to x1 + 7x2 ≥ 2 2x1 + x2 ≥ 7 7x1 + 9x2 ≥ 1 x1, x2 ≥ 0 A) Min C = 7 at x1 = 21 , x2 = 0 2 2

B) Min C = 21 at x1 = 0, x2 = 7 2 2

C) Min C = 21 at x1 = 7 , x2 = 0 2

D) Min C = 21 at x1 = 7 , x2 = 0 2 2

Refer to the given system to solve the problem. 29) Find the solution of the system for which x2 = 0 and s 1 = 0

29)

5x1 + 3x2 + s 1 = 90 2x1 + 4x2 + s 2 = 64 A) (x1, x2, s 1, s 2) = (16, 0, 0, 42)

B) (x1, x2, s 1, s 2) = (90, 0, 0, 64)

C) (x1, x2, s 1, s 2) = (18, 0, 0, 28)

D) (x1, x2, s 1, s 2) = (32, 0, 0, 45)

Pivot once about the circled element in the simplex tableau, and read the solution from the result. 30)

A) x3 = 5, x2 = -7, z = 10; x1, s 1, s 2 = 0

B) x3 = 5, x2 = -7, z = -10; x1, s 1, s 2 = 0

C) x3 = 10, x2 = 10, z = 7; x1, s 1, s 2 = 0

D) x3 = 10, x2 = 7, z = 10; x1, s 1, s 2 = 0

30)

Graph the system of inequalities. 31) x1 + x2 ≤ 40 3x1 + x2 ≤ 84 x1, x2 ≥ 0

31)

(13, 45)

(0, 40)

(22, 18)

25 20

(22, 18)

(40, 0) 5 10 15 20 25 30

(28, 0) 5

(13, 45)

C) x2

x2 35

(22, 18)

10 5

10 15 20 25 30 35 40 x1

(22, 18)

10 5

(28, 0) 5 10 15 20 25 30

(40, 0) 5 10 15 20 25 30

Solve the given linear programming problem using the table method. 32) Maximize P = 7x1 + 6 x2 subject to

(40, 0)

3x1 + x2 ≤ 21 x1 + x2 ≤ 10 x1 + 2x2 ≤ 12 x1, x2 ≥ 0

A) Max P = 60 at x1 = 6, x2 = 3

B) Max P = 66 at x1 = 6, x2 = 4

C) Max P = 68 at x1 = 8, x2 = 2

D) Max P =65.5 at x1 = 5.5, x2 = 4.5

32)

Provide an appropriate response. 33) Write the basic solution for the following simplex tableau:

33)

x1 x2 x3 s 1 s 2 s 3 P 0 10 7 0 1 1 0 12 1 -7 6 0 0 1 0 28 0 0 10 1 0 0 0 30 0 -5 -8 0 0 4 1 50 A) x1 = 28, x2 = 0, x3 = 0, s 1 = 30, s 2 = 12, s 3 = 0, P = - 50 B) x1 = 28, x2 = 0, x3 = 0, s 1 = 30, s 2 = 12, s 3 = 0, P = 50 C) x1 = 28, x2 = 10, x3 = 7, s 1 = 30, s 2 = 12, s 3 = 0, P = 50 D) x1 = 28, x2 = 10, x3 = 0, s 1 = 30, s 2 = 12, s 3 = 0, P = 50 34) Write the simplex tableau, label the columns and rows, underline the pivot element, and identify the entering and exiting variables for the linear programming problem: Maximize

P = 5x1 + 3x2

subject to 3x1 + 8x2 ≤ 5 3x1 + 5x2 ≤ 6 -8x1 + 6x2 ≤ 32 6x2 ≤ 3 x1, x2 ≥ 0 Enter

x1 3 3 -8 0 -5

s1 s Exit 2 s3 s4 P

x2 8 5 6 6 -3

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

x2 8 5 6 6 -3

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

1 0 0 1 0

0 0 0 0 1

5 6 32 3 0

Exit s1 Enter

s2 s3 s4 P

x1 3 3 -8 0 -5

5 6 32 3 32

34)

Exit x1 3 3 -8 0 -5

s1 s2

Enter

s3 s4 P

x2 8 5 6 6 -3

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 1 0 1

x2 8 5 6 6 -3

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

1 0 0 1 0

0 0 0 0 1

6 6 32 3 0

Enter s1 Exit

s2 s3 s4 P

x1 3 3 -8 0 -5

8 6 32 3 0

35) Use the big M method to find the optimal solution to the problem. Maximize P = 3 x1 + 6 x2 + 2x3 subject to

2x1 + 2x2 + 3 x3 ≤ 12 2x1 + 2x2 + x3 = 0

x1, x2, x3 ≥ 0 A) Max P = 27 at x1 = 0, x2 = 3, x3 = 3

B) Max P = 27 at x1 = 3, x2 = 3, x3 = 3

C) Max P = 27 at x1 = 3, x2 = 3, x3 = 0

D) Max P = -27 at x1 = 3, x2 = 3, x3 = 3

35)

36) Find the initial simplex tableau for the following preliminary simplex tableau (DO NOT SOLVE): x1 x2 s 1 s 2 s 3 a1 a2 P 2 1 -1 0 0 1 0 0 7 1 3 0 -1 0 0 1 0 12 1 1 0 0 1 0 0 0 5 -4 -9 0 0 0 M M 1 0 A) x1 x2 s 1 s 2 s 3 a1 a2 P 2 1 -1 0 0 1 0 0 7 a2 1 3 0 -1 0 0 1 0 12 s3 1 1 0 0 1 0 0 0 5 -3M-4 -4M-9 M M 0 0 0 1 -19M P a1

B) x1 x2 s 1 s 2 s 3 a1 a2 P 2 1 -1 0 0 1 0 0 -7 a2 1 3 0 -1 0 0 1 0 -12 s3 1 1 0 0 1 0 0 0 -5 P 3M-4 4M-9 M M 0 0 0 -1 19M a1

C) x1 x2 s 1 s 2 s 3 a1 a2 P 2 1 -1 0 0 1 0 0 7 a2 1 3 0 -1 0 0 1 0 12 s3 1 1 0 0 1 0 0 0 5 3M-4 4M-9 M M 0 0 0 -1 19M P a1

D) x1 x2 s 1 s 2 s 3 a1 a2 P 2 1 -1 0 0 1 0 0 7 a2 1 3 0 -1 0 0 1 0 12 s3 1 1 0 0 1 0 0 0 5 P 3M-4 4M-9 M M 0 0 0 1 -19M a1

36)

Introduce slack variables as necessary, and write the initial simplex tableau for the problem. 37) Find x1 ≥ 0 and x2 ≥ 0 such that

37)

2x1 + 5 x2 ≤ 16 3x1 + 3 x2 ≤ 4 and z = 4x1 + x2 is maximized.

x1 x2 s 1 s 2 2 5 1 0 3 3 0 1 4 1 0 0 x1 x2 s 1 s 2 2 5 1 0 3 3 0 1 4 1 0 0

x1 x2 s 1 s 2 2 5 1 0 3 3 0 1 -4 -1 0 0 D) x1 x2 s 1 s 2 2 5 1 0 3 3 0 1 -4 -1 0 0 B)

0 0 1 z

16 4 0

0 0 1

4 16 0

z 0 0 1 z

4 16 0

0 0 1

16 4 0

Provide an appropriate response. 38) Solve the following linear programming problem using the simplex method: Maximize P = x1 - x2

38)

subject to x1 + x2 ≤ 4 2x1 + 7x2 ≤ 14 x1, x2 ≥ 0 A) Max P = 14 at x1= 4 and x2 = 0

B) Max P = 4 at x1= 4 and x2 = 0

C) Max P = 4 at x1= 14 and x2 = 0

D) Max P = 4 at x1= 4 and x2 = 4

Solve the problem. 39) A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average monthly salary of a counselor is $2400 and the average monthly salary of an aide is $1100. The camp can accommodate up to 45 staff members and needs at least 30 to run properly. They must have at least 10 aides, and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the camp hire to minimize cost? A) 27 counselors and 18 aides B) 12 counselors and 18 aides C) 35 counselors and 10 aides D) 18 counselors and 12 aides Find the transpose of the matrix. 40) 8 2 0 9 1 7 8 6 A) 1 7 8 6 8 2 0 9

39)

40)

C) 3 2 2 1 9 6 0 8

D) 8 1 2 7 0 8 9 6

1 8 7 2 8 0 6 9

Solve the given linear programming problem using the table method. 41) Maximize P = 6 x1 + 7x2 subject to

41)

2x1 + 3 x2 ≤ 12 2x1 + x2 ≤ 8

x1, x2 ≥ 0 A) Max P = 32 at x1 = 3, x2 = 2

B) Max P = 55 at x1 = 4, x2 = 4

C) Max P = 32 at x1 = 2, x2 = 3

D) Max P = 24 at x1 = 4, x2 = 0

Graph the system of inequalities. 42) x1 + x2 ≤ 12

42)

2x1 + x2 ≤ 20 x1, x2 ≥ 0 A)

B) 22 20

(0, 20)

14 12

x2 (0, 20)

14 (0, 12)

(8, 4)

4 2 4

(8, 4)

4 2

(10, 0) (12, 0) 2

(0, 12)

8 10 12 14 16 18 20 x1

(10, 0) (12, 0) 2

8 10 12 14 16 18 20 x1

D) 22 20

(0, 20)

14 12

x2 (0, 20)

14 (0, 12)

(8, 4)

4 2 4

(8, 4)

4 2

(10, 0) (12, 0) 2

(0, 12)

8 10 12 14 16 18 20 x1

(10, 0) (12, 0) 2

8 10 12 14 16 18 20 x1

Introduce slack variables as necessary, and write the initial simplex tableau for the problem. 43) Find x1 ≥ 0 and x2 ≥ 0 such that

43)

x1 + x2 ≤ 90 3x1 + x2 ≤ 152 and z = 2x1 + x2 is maximized.

x1 x2 s 1 s 2 1 1 1 0 3 1 0 1 2 1 0 0 x x C) 1 2 s1 s2 1 1 1 0 3 1 0 1 -2 -1 0 0

x1 x2 s 1 s 2 1 1 1 0 3 1 0 1 2 1 0 0 x x D) 1 2 s1 s2 1 1 1 0 3 1 0 1 -2 -1 0 0 B)

0 0 1 z

90 152 0

0 0 1

152 90 0

z 0 0 1 z

152 90 0

0 0 1

90 152 0

Provide an appropriate response. 44) Write the modified problem for the following linear programming problem (DO NOT SOLVE): Maximize

P = 9x1 - 4x2 + x3

subject to 4x1 - x2 + 2x3 ≤ 3 -x1 - 7x2 + 9x3 ≤ -4 x1 - x2 + 2x3 = 8 x1, x2, x3 ≥ 0 A) Maximize

P = 9x1 - 4x2 + x3 + Ma1+ Ma2

subject to 4x1 - x2 + 2x3 + s 1 = -3 x1 + 7x2 - 9x3 - s 2 + a1 = -4 x1 - x2 + 2x3 + a2 = - 8 x1, x2, x3 , s 1, s 2, a1, a2 ≥ 0 B) Maximize

P = 9x1 - 4x2 + x3 - Ma1 - Ma2

subject to 4x1 - x2 + 2x3 + s 1 = -3 x1 + 7x2 - 9x3 - s 2 + a1 = -4 x1 - x2 + 2x3 + a2 = - 8 x1, x2, x3 , s 1, s 2, a1, a2 ≥ 0 C) Maximize

P = 9x1 - 4x2 + x3 + Ma1 + Ma2

subject to 4x1 - x2 + 2x3 + s 1 = 3 x1 + 7x2 - 9x3 - s 2 + a1 = 4 x1 - x2 + 2x3 + a2 = 8 x1, x2, x3 , s 1, s 2, a1, a2 ≤ 0 18

44)

D) Maximize

P = 9x1 - 4x2 + x3 - Ma1 - Ma2

subject to 4x1 - x2 + 2x3 + s 1 = 3 x1 + 7x2 - 9x3 - s 2 + a1 = 4 x1 - x2 + 2x3 + a2 = 8 x1, x2, x3 , s 1, s 2, a1, a2 ≥ 0 Find the transpose of the matrix. 45) 2 6 3 2 1 3 9 8 7 A) 6 3 2 1 3 2 8 7 9

45)

C) 2 2 9 6 1 8 3 3 7

D) 2 1 2 2 6 3 9 8 7

9 8 7 2 1 3 2 6 3

Solve the problem. 46) Formulate the following problem as a linear programming problem (DO NOT SOLVE):A veterinarian wants to set up a special diet that will contain at least 500 units of vitamin B1 at least 800 units of vitamin B2 and at least 700 units of vitamin B6 . She also wants to limit the diet to at most 300 total grams. There are three feed mixes available, mix P, mix Q, and mix R. A gram of mix P contains 3 units of vitamin B1, 5 units of vitamin B2, and 8 units of vitamin B6 . A gram of mix Q contains 9 units of vitamin B1, 8 units of vitamin B2, and 6 units of vitamin B6 . A gram of mix R contains 7 units of vitamin B1, 6 units of vitamin B2, and 9 units of vitamin B6 . Mix P costs $0.10 per gram, mix Q costs $0.12 per gram, and mix R costs $0.21 per gram. How many grams of each mix should the veterinarian use to satisfy the requirements of the diet at minimal costs? (Let x1 equal the number of grams of mix P, x2 equal the number of grams of mix Q, and x3 equal the number of grams of mix R that are used in the diet). A) Minimize C = 0.10x 1 + 0.12x 2 + 0.21x 3 subject to 3x1 + 9x2 + 7x3 ≥ 500 5x1 + 8x2 + 6x3 ≥ 800 8x1 + 6x2 + 9x3 ≥ 700 x1 + x2 + x3 ≤ 300 B) Minimize

x1, x2, x3 ≥ 0 C = -0.10x 1- 0.12x 2 - 0.21x 3

subject to 3x1 + 9x2 + 7x3 ≤ 500 5x1 + 8x2 + 6x3 ≤ 800 8x1 + 6x2 + 9x3 ≥ 700 x1 + x2 + x3 ≤ 300 x1, x2, x3 ≥ 0

46)

C) Minimize

C = 0.10x 1- 0.12x 2 - 0.21x 3

subject to 3x1 + 9x2 + 7x3 ≤ 500 5x1 + 8x2 + 6x3 ≤ 800 8x1 + 6x2 + 9x3 ≥ 700 x1 + x2 + x3 ≤ 300 D) Minimize

x1, x2, x3 ≥ 0 C = 0.10x 1- 0.12x 2 + 0.21x 3

subject to 3x1 + 9x2 + 7x3 ≤ 500 5x1 + 8x2 + 6x3 ≥ 800 8x1 + 6x2 + 9x3 ≥ 700 x1 + x2 + x3 ≤ 300 x1, x2, x3 ≥ 0 Provide an appropriate response. 47) Solve the following linear programming problem using the simplex method: Maximize P = 5 x1 + 3 x2 subject to 2x1 + 4x2 ≤ 13 x1 + 2x2 ≤ 6 x1 , x2 ≥ 0 A) Max P = 32.5 when x1 = 6.5, x2 = 0

B) Maxi P = 9 when x1 = 0, x2 = 3

C) Max P = 18 when x1 = 0, x2 = 6

D) Max P = 30 when x1 = 6, x2 = 0

47)

48)

= 14

B) x1 + 2x2 + s 1 ≤ 14 4x1 + x2 + s 2 ≤ 16

4x1 + x2 + s 2 = 16 x1 x2 s 1 s 2

x1 x2 s1 s2 (A) 0 0 14 16 (B) 0 7 0 9 (C) 0 16 -18 0 (D) 14 0 0 -40 (E) 4 0 10 0 18 - 16 (F) 0 0 7 7

(A) 0 0 16 14 (B) 0 7 0 -9 (C) 0 16 18 0 (D) 14 0 0 40 (E) 4 0 -10 0 40 18 (F) 0 0 7 7 Feasible: (A), (C), (D), (F) Not feasible: (B), (E) C) x1 + 2x2 + s 1 = 14

Feasible: (A), (B), (E) Not feasible: (C), (D), (F) D) x1 + 2x2 + s 1 = 14

4x1 + x2 + s 2 = 16 x1 x2 s 1 s 2

(A) 0 0 14 16 (B) 0 7 0 9 (C) 0 16 -18 0 (D) 14 0 0 -40 (E) 4 0 10 0 18 40 (F) 0 0 7 7

(A) 0 0 16 14 (B) 0 7 0 9 (C) 0 16 -18 0 (D) 14 0 0 -40 (E) 4 0 10 0 40 18 (F) 0 0 7 7

Feasible: (A), (B), (E), (F) Not feasible: (C), (D)

Provide an appropriate response. 49) Solve the following linear programming problem using the simplex method: Maximize P = 7x1 + 2x2 + x3 subject to: x1 + 5 x2 + 7x3 ≤ 8 x1 + 4x2 + 11x3 ≤ 9 x1 , x2 , x3 ≥ 0 A) Max P = 0 when x1 = 0, x2 = 0, x3 = 8

B) Max P = 9 when x1 = 1, x2 = 1, x3 = 0

C) Max P = 56 when x1 = 8, x2 = 0, x3 = 0

D) Max P = 63 when x1 = 9, x2 = 0, x3 = 0

49)

50) Convert the inequality to a linear equation by adding a slack variable. 9x1 + x2 + 4x3 ≤ 240 A) 9x1 + x2 + 4x3 + s 1 ≤ 240

B) 9x1 + x2 + 4x3 + s 1 ≥ 240

C) 9x1 + x2 + 4x3 + s 1 = 240

D) 9x1 + x2 + 4x3 + s 1 + 240 = 0

Refer to the table of the six basic solutions to the e-system. 51) x1 + 2x2 + s 1 = 14 3x1 + 4x2

50)

51)

+ s 2 = 36

x1 x2 s 1 s 2 (A) 0 0 14 36 (B) 0 7 0 8 (C) 0 9 -4 0 (D) 14 0 0 -6 (E) 12 0 2 0 (F) 8 3 0 0 Which of the six basic solutions are feasible? Which are not feasible? Use the basic feasible solutions to find the maximum value of P = 24x1 + 18x2. A) Feasible: (A), (E), (F); Not feasible: (C), (D), (B); The maximum value of P is 288. B) Feasible: (A), (B), (E), (F); Not feasible: (C), (D); The maximum value of P is 246. C) Feasible: (A), (B), (E), (F); Not feasible: (C), (D); The maximum value of P is 126. D) Feasible: (A), (B), (E), (F); Not feasible: (C), (D); The maximum value of P is 288. Solve the problem. 52) A catering company makes three kinds of Jello saladorange, strawberry, and lemon. It supplies two outlet stores, Outlet A and Outlet B. The company can make up to 14 trays of orange, 20 trays of strawberry, and 24 trays of lemon per day. Outlet A needs at least 20 trays of Jello per day and Outlet B needs at least 10 trays of Jello per day. The transportation costs for shipping trays from the company to the outlets are given in the chart below:

Type of Jello Orange Strawberry Lemon

Shipping Charges Outlet A Outlet B $4 $1 $1 $4 $5 $5

Formulate a linear programming problem that will determine a shipping schedule that will minimize the cost of transporting the Jello salads to fill the needs of the two outlets. DO NOT SOLVE.

52)

Let

x1 = number of trays of orange shipped to Outlet A x2 = number of trays of strawberry shipped to Outlet A x3 = number of trays of lemon shipped to Outlet A x4 = number of trays of orange shipped to Outlet B x5 = number of trays of strawberry shipped to Outlet B x6 = number of trays of lemon shipped to Outlet B

A) Minimize

C = -4x1 - x2 - 5x3 - x4 + 4x5 - 5x6

subject to x1 + x4 ≥ 14 x2 + x5 ≥ 20 x3 + x6 ≤ 24 x1 + x2 + x3 ≥ 20 x4 + x5 + x6 ≥ 10 x1, x2, x3 , x4, x5 , x6 ≥ 0 B) Minimize

C = -4x1 - x2 + 5x3 + x4 + 4x5 - 5x6

subject to x1 + x4 ≤ 14 x2 + x5 ≤ 20 x3 + x6 ≤ 24 x1 + x2 + x3 ≥ 20 x4 + x5 + x6 ≥ 10 x1, x2, x3 , x4, x5 , x6 ≥ 0 C) Minimize

C = -4x1 - x2 + 5x3 + x4 + 4x5 - 5x6

subject to x1 + x4 ≥ 14 x2 + x5 ≥ 20 x3 + x6 ≤ 24 x1 + x2 + x3 ≥ 20 x4 + x5 + x6 ≥ 10 x1, x2, x3 , x4, x5 , x6 ≥ 0 D) Minimize

C = 4x1 + x2 + 5x3 + x4 + 4x5 + 5x6

subject to x1 + x4 ≤ 14 x2 + x5 ≤ 20 x3 + x6 ≤ 24 x1 + x2 + x3 ≥ 20 x4 + x5 + x6 ≥ 10 x1, x2, x3 , x4, x5 , x6 ≥ 0

Provide an appropriate response. 53) Write the basic solution for the following simplex tableau: x1 x 2 x3 s 1 s 2 P 3 4 0 3 1 0 20 1 5 1 7 0 0 28 -3 4 0 1 0 1 20 A) x1, x2, s 1 = 0, x5 = 28, s 2 = 20, P = 20

B) x1, x2, s 1 = 0, x3 = 20, s 2 = 28, P = 20

C) x1, x2, s 1 = 0, x3 = 28, s 2 = 20, P = 20

D) x1, x2, s 1 = 0, x1 = 28, s 2 = 20, P = 20

54) Solve the linear programming problem by using the simplex method. Minimize f = 5 x + 3y subject to: 2x + 3y ≥ 9 2x + y ≥ 11 x ≥ 0, y ≥ 0 A) x = 11 , y = 0, f = 55 2 2

53)

54)

B) x = 0, y = 11, f = 33

C) x = 5, y = 1, f = 28

D) x = 5, y = 0, f = 25

Solve the problem. 55) A linear programming problem has 35 decision variables x1, ..., x35 and50 problem constraints.

55)

How many rows are there in the table of basic solutions of the associated e -system? A) 8.96 × 10 25 B) 8.96 × 10 23 C) 8.96 × 10 13 D) 2.25 × 10 23 Write the e-system obtained via slack variables for the linear programming problem. 56) Maximize P = 2x1 + 8 x2

56)

subject to: x1 + 3 x2 ≤ 15 8x1 + 3 x2 ≤ 25 with:

x1 ≥ 0, x2 ≥ 0 A) x1 + 3 x2 + s 1 = 15

B) x1 + 3 x2 = s 1 + 15

8x1 + 3 x2 + s 1 = 25 C) x1 + 3 x2 + s 1 ≤ 15

8x1 + 3 x2 = s 2 + 25 D) x1 + 3 x2 + s 1 = 15

8x1 + 3 x2 + s 2 ≤ 25

8x1 + 3 x2 + s 2 = 25

Provide an appropriate response. 57) Maximize P= x1 + 2x2 + 3 x3

57)

subject to: x1 + 8 x2 + 5 x3 ≤ 40 9x1 + x2 + 7x3 ≤ 50 with:

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 How many slack variables must be introduced to form the system of problem constraint equations? A) 1 B) 0 C) 3 D) 2

58) Solve the linear programming problem. Minimize C = 10 x1 + 12x2 + 28 x3 subject to

58)

4x1 + 2x2 + 3 x3 ≥ 20 3 x1 - x2 - 4x3 ≤ 10

x1, x2, x3 ≥ 0 A) Min C = 64 at x1 = 4, x2 = 2, x3 = 0

B) Min C = 64 at x1 = 0, x2 = 2, x3 = 2

C) Min C = 64 at x1 = 0, x2 = 2, x3 = 4

D) Min C = 64 at x1 = 4, x2 = 4, x3 = 2

Answer Key Testname: CHAP 06_14E

1) (A) Maximize

P = 3x1 + 7x2 - Ma1 - Ma2

subject to x1 + x2 - s 1 + a1 = 6 2x1 + x2 - s 2 + a2 = 5 x1 + 3x2 + s 3 = 15 x1, x2, s 1, s 2, s 3 , a1, a2 ≥ 0 (B) x1 x2 s 1 s 2 s 3 a1 a2 P 1 1 -1 0 0 1 0 0 6 2 1 0 -1 0 0 1 0 5 1 3 0 0 1 0 0 0 15 -3 -7 0 0 0 M M 1 0 2) 100 of type -I, 150 of type -II, and none of type -III systems; maximum profit = $6500 3) Min C = 170 at u =1 and v = 2 4) Basic: s 1, s 2, P; nonbasic: x, y; y

3 5 (4) 1 -15 -10

1 0 0

0 1 0

0 0 1

17 4

-3 4

1 4

0 -

25 4

15 4

s1 s2 P

15 Enter x, Exit s 2 4 0

5) Optimal solution has been found. 6) (a) 7 Holiday boxes, 5 Noel boxes; , Maximum profit = $82 (b) none 7) Min C = 21 at u =1 and v = 3 8) There is no solution. 9) A 10) D 11) C 12) C 13) D 14) B 15) A 16) D 17) D 26

Answer Key Testname: CHAP 06_14E

18) B 19) C 20) C 21) D 22) C 23) E 24) A 25) A 26) B 27) C 28) D 29) C 30) B 31) B 32) A 33) B 34) A 35) C 36) A 37) D 38) B 39) B 40) C 41) A 42) D 43) D 44) D 45) B 46) A 47) D 48) C 49) C 50) C 51) D 52) D 53) C 54) A 55) B 56) D 57) C 58) A

CHAPTER 7

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Which of the following is NOT a subset of the set {p, o, 7}? A) {o, 7} B) 7 C) {p, o, 7}

1) D) ∅

2) One of the following is false; indicate by letter which one: A) 5 ∈ {3, 4, 5, 6} B) 4 ⊂ {3, 4, 5, 6} C) 4 ∈ {3, 4, 5, 6} D) {4} ⊂ {3, 4, 5, 6}

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) A person purchasing a new car has several options: 6 interior color choices, 5 exterior color choices, 2 choices of radios, and 5 choices of body styles. How many different cars are possible if one choice is made for each option?

4) A combination lock on a suitcase has 5 wheels, each labeled with digits 1 to 8. How many 5 -digit combination lock codes are possible if no digit can be repeated?

5) Construct a truth table to determine whether or not the following implication is true: ~(p ∨ q) ≡ ~p ∧ ~q.

6) How many nine -digit ZIP code numbers are possible if the first digit cannot be a four and adjacent digits cannot be the same?

Use a Venn Diagram and the given information to determine the number of elements in the indicated region. ′ ′ 7) Let U = {a, l, i, t, e}, A = {l, i, t},B = {l, e}, C = {a, l, i, t, e}, and D = {a, e}. Find (C ∩ B ) ∪ A . 7) Provide an appropriate response. 8) How many different five -letter code words are possible from the first ten letters of the alphabet if the first letter cannot be a vowel and adjacent letters must be different.

9) State the converse and contrapositive of the position, "If n is an integer that is a multiple of 15, then n is an integer that is a multiple of 3 and a multiple of 5."

10) Construct a truth table for the proposition and determine whether it is a contingency, a tautology, or a contradiction: ~p ∨ q.

10)

Use a Venn Diagram and the given information to determine the number of elements in the indicated region. 11) In a marketing survey involving 1,000 randomly chosen people, it is found that 660 use 11) brand P, 440 use brand Q, and 220 use both brands. How many people in the survey use brand P and not brand Q?

Provide an appropriate response. 12) A test is composed of 4 multiple choice problems and 8 questions that can be answered true or false. Each multiple choice problem has 4 choices. How many different response sheets are possible if only one choice is marked for each question?

12)

13) A coin that can turn up either heads (H) or tails (T) is flipped. If a head turns up on the first toss, a spinner that can land on any of the first 7 natural numbers is spun. If a tail turns up, the coin is flipped a second time. What are the different possible outcomes?

13)

14) Construct a truth table for the proposition and determine whether it is a contingency, a tautology, or a contradiction: (q ∨ p) ∧ ~q.

14)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Tell whether the statement is true or false. 15) 3 ∈ {6, 9, 12, 15, 18} A) True

15) B) False

Provide an appropriate response. 16) Suppose there are 4 trains connecting town X to town Y and 6 roads connecting town Y to town Z. In how many ways can a person travel from X to Z via Y? A) 12 B) 16 C) 10 D) 24 E) 36

16)

Determine whether the given set is disjoint or not disjoint. Consider the set N of positive integers to be the universal set, and let A = {n ∈ N| n > 50} B = {n ∈ N| n < 250} O = {n ∈ N| n is odd} E = {n ∈ N| n is even} 17) A' ∩ B' 17) A) not disjoint B) disjoint Determine whether the selection is a permutation, a combination, or neither. 18) An animal trainer selects 2 of the 5 baboons to showcase on the talk show. A) permutation B) combination C) neither

18)

Determine whether the given set is finite or infinite. Consider the set N of positive integers to be the universal set, and let A = {n ∈ N| n > 50} B = {n ∈ N| n < 250} O = {n ∈ N| n is odd} E = {n ∈ N| n is even} 19) A ∩ O' 19) A) finite B) infinite Construct a truth table to decide if the two statements are equivalent. 20) ~(~q); q A) True B) False

20)

Provide an appropriate response. 21) Results of a survey of fifty students indicate that 30 like red jelly beans, 29 like green jelly beans, and 17 like both red and green jelly beans. How many of the students surveyed like neither red nor green jelly beans? A) 8 B) 12 C) 13 D) 17 22) The access code to a house's security system consists of five digits. How many different codes are available if each digit can be repeated? A) 45 B) 5 C) 100,000 D) 3125 E) 32 Tell whether the statement is true or false. 23) 0 ∉ ∅ A) True

21)

22)

23) B) False

Use a Venn Diagram and the given information to determine the number of elements in the indicated region. 24) At Southern States University (SSU) there are 399 students taking Finite Mathematics or 24) Statistics. 238 are taking Finite Mathematics, 184 are taking Statistics, and 23 are taking both Finite Mathematics and Statistics. How many are taking Finite Mathematics but not Statistics? A) 161 B) 215 C) 192 D) 376 Solve the problem. 25) In a Power Ball lottery, 5 numbers between 1 and 12 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is not important. A) 95,040 B) 120 C) 248,832 D) 792

25)

Construct a truth table to decide if the two statements are equivalent. 26) q → p ; p → q A) True B) False

26)

Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false. 27) {9} ⊂ A A) True B) False

27)

Provide an appropriate response. 28) Mrs. Bollo's second grade class of thirty students conducted a pet ownership survey. Results of the survey indicate that 8 students own a cat, 15 students own a dog, and 5 students own both a cat and a dog. How many of the students surveyed own no dogs? A) 15 B) 3 C) 20 D) 8

28)

Construct a truth table for the proposition. 29) p → ~q A) p q p → ~q T T F T F F F T T F F T C) p q p → ~q T T T T F T F T F F F F

29) B) p T T F F

q p → ~q T T F F T T F T

p T T F F

q p → ~q T F F T T T F T

Solve the problem. 30) In how many ways can a student work 7 out of 10 questions on an exam? A) 10,000,000 B) 120 C) 21

30) D) 720

Use the Venn diagram below to find the number of elements in the region.

31) n(A ∪ B) A) 14

B) 21

C) 29

D) 11

31)

32) n((A ∪ B) ∩ C) A) 14

B) 33

C) 11

D) 15

32)

Express the proposition as an English sentence and determine whether it is true or false, where p and q are the propositions p: "9 ∙ 9 = 81" q: "8 ∙ 10 < 7 ∙ 11 33) The contrapositive of p → q 33) A) If 8 ∙ 10 is not less than 7 ∙ 11, then 9 ∙ 9 is not equal to 81; false B) If 8 ∙ 10 is less than 7 ∙ 11, then 9 ∙ 9 is equal to 81; false C) If 8 ∙ 10 is less than 7 ∙ 11, then 9 ∙ 9 is not equal to 81; false D) If 8 ∙ 10 is not less than 7 ∙ 11, then 9 ∙ 9 is equal to 81; false Tell whether the statement is true or false. 34) {all odd integers greater than -3 and less than 5} = {-1, 1, 3} A) True B) False

34)

Express the proposition as an English sentence and determine whether it is true or false, where p and q are the propositions p: "9 ∙ 9 = 81" q: "8 ∙ 10 < 7 ∙ 11 35) p ∧ q 35) A) If 9 ∙ 9 = 81, then 8 ∙ 10 < 7 ∙ 11; false B) 9 ∙ 9 = 81 or 8 ∙ 10 < 7 ∙ 11; true C) 8 ∙ 10 is not less than 7 ∙ 11; true D) 9 ∙ 9 = 81 and 8 ∙ 10 < 7 ∙ 11; false Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false. 36) {3, 0, 8} ∈ A A) True B) False

36)

Determine whether the given set is finite or infinite. Consider the set N of positive integers to be the universal set, and let A = {n ∈ N| n > 50} B = {n ∈ N| n < 250} O = {n ∈ N| n is odd} E = {n ∈ N| n is even} 37) B ∪ E 37) A) infinite B) finite Solve the problem. 38) A home cooking equipment supply company employs 7 sales representatives and 6 product designers. How many ways can this company select 4 of these employees to send to a product demonstration convention in Oahu if at least 3 product designers must attend? A) 42 B) 155 C) 504 D) 168

38)

Determine whether the given set is disjoint or not disjoint. Consider the set N of positive integers to be the universal set, and let A = {n ∈ N| n > 50} B = {n ∈ N| n < 250} O = {n ∈ N| n is odd} E = {n ∈ N| n is even} 39) O ∩ E' 39) A) not disjoint B) disjoint Construct a truth table to decide if the two statements are equivalent. 40) ~p ∨ ~q; ~(p ∧ q) A) True B) False

40)

Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false. 41) {5} ⊆ D 41) A) True B) False Provide an appropriate response. 42) In a group of 42 students, 22 take history, 17 take biology and 8 take both history and biology. How many students take neither biology nor history? A) 22 B) 5 C) 8 D) 11

42)

Use the addition principle for counting to solve the problem. 43) If n(A) = 20, n(A ∪ B) = 58, and n(A ∩ B) = 16, find n(B). A) 55 B) 58 C) 53

43) D) 54

Provide an appropriate response. 44) A survey of residents in a certain town indicates 170 own a dehumidifier, 130 own a snow blower, and 80 own a dehumidifier and a snow blower. How many own a dehumidifier or a snow blower? A) 220 B) 250 C) 170 D) 80 Tell whether the statement is true or false. 45) 7 ∉ {14, 21, 28, 35, 42} A) True

44)

45) B) False

Construct a truth table for the proposition. 46) ~p → (~p ∧ s) A) p s ~p →(~p ∧ s) T T T T F T F T T F F F C) p s ~p →(~p ∧ s) T T F T F F F T T F F F

46) B) p T T F F

s ~p →(~p ∧ s) T T F T T T F T

p T T F F

s ~p →(~p ∧ s) T T F F T T F F

Solve the problem. 47) How many ways can a committee of 6 be selected from a club with 10 members? A) 210 B) 1,000,000 C) 151,200 D) 60

47)

Use the addition principle for counting to solve the problem. 48) If n(A) = 5, n(B) = 11 and n(A ∩ B) = 3, what is n(A ∪ B)? A) 14 B) 12 C) 13

48)

Construct a truth table for the proposition. 49) p ∨ (p ∧ ~p) A) p p ∨ (p ∧ ~p) T T F T C) p p ∨ (p ∧ ~p) T T F F

D) 11

49) B) p p ∨ (p ∧ ~p) T F F T D) p p ∨ (p ∧ ~p) T F F F

Use the addition principle for counting to solve the problem. 50) If n(B) = 24, n(A ∩ B) =5, and n(A ∪ B) = 42, find n(A). A) 23 B) 24 C) 25

50) D) 21

Evaluate. 51) P 10, 2

51)

A) 45

B) 8

C) 90

D) 19

Solve the problem. 52) A software company employs 9 sales representatives and 8 technical representatives. How many ways can the company select 5 of these employees to send to a computer convention if at least 4 technical representatives must attend the convention? A) 180 B) 360 C) 1440 D) 686 Evaluate. 53) C8, 2

52)

53)

A) 28

B) 4

C) 720

D) 1440

Solve the problem. 54) How many 4-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, if repetition of digits is not allowed? A) 23 B) 24 C) 2401 D) 840 Use the Venn diagram to find the requested set. 55) Find A ∪ B.

54)

55)

q A) {e , b, j, h, q, m, t} C) {q}

B) {b, j} D) {e , b, j, h, m, t}

Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false. 56) U ⊆ A 56) A) True B) False Use a Venn Diagram and the given information to determine the number of elements in the indicated region. 57) n(A) = 33, n(B) = 15, n(A ∪ B) = 42, n(B') = 40. Find n(A ∩ B)'. 57) A) 13 B) 42 C) 36 D) 49 Provide an appropriate response. 58) In a group of 42 students, 22 take history, 17 take biology and 8 take both history and biology. How many students take biology, but not history? A) 22 B) 17 C) 5 D) 9

58)

Evaluate. 59)

7! 5!

59) A) 42

B) 7

7 5

D) 2!

Use the Venn diagram below to find the number of elements in the region.

60) n(A ∩ B ∩ C) A) 16

60) B) 44

C) 18

D) 8

Provide an appropriate response. 61) A restaurant offered pizza with 3 types of crusts and 7 different toppings. How many different types of pizzas could be offered? A) 10 B) 9 C) 21 D) 63 E) 49

61)

Use the Venn diagram below to find the number of elements in the region.

′

62) n(C ) A) 14

62) B) 29

C) 39

D) 24

B) 6

C) 12

D) 48

Evaluate. 63) 4!

63) A) 24

Use the addition principle for counting to solve the problem. 64) If n(A) = 40, n(B) = 117 and n(A ∪ B) = 137, what is n(A ∩ B)? A) 10 B) 20 C) 40

64) D) 22

Provide an appropriate response. 65) In Virginia, each automobile license plate consists of a single digit followed by three letters, followed by three digits. How many distinct license plates can be formed if there are no restrictions on the digits or letters? A) 175,760 B) 17,576 C) 175,7560,000 D) 17,575,600 E) 10,757,600 66) How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed. A) 5000 B) 2000 C) 2997 D) 1512 E) 2999 Use the Venn diagram to find the requested set. 67) Find A ∩ B.

65)

66)

67)

m A) {m} C) {e , c, i, g, m, q, t}

B) {c, i} D) {e , c, i, g, q, t}

Solve the problem. 68) How many ways can a committee of 4 be selected from a club with 12 members? A) 248 B) 24 C) 11,880 D) 495

68)

Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false. 69) {3, 0, 8} ⊂ A A) True B) False

69)

Determine whether the given set is finite or infinite. Consider the set N of positive integers to be the universal set, and let A = {n ∈ N| n > 50} B = {n ∈ N| n < 250} O = {n ∈ N| n is odd} E = {n ∈ N| n is even} 70) A' 70) A) infinite B) finite Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false. 71) C ⊆ D 71) A) True B) False

Solve the problem. 72) A pollster wants to minimize the effect the order of the questions has on a person's response to a survey. How many different surveys are required to cover all possible arrangements if there are 6 questions on the survey? A) 36 B) 6 C) 120 D) 720 Use the Venn diagram to find the requested set. 73) Find A.

72)

73)

6 2

A) {7, 2, 6}

B) {6, z, q, h}

C) {7, 2, 6, z}

D) {6}

Determine whether the selection is a permutation, a combination, or neither. 74) Mary baked 3 pies: 1 for her father, 1 for her friend Joe, and 1 for her coworkers. A) permutation B) neither C) combination Provide an appropriate response. 75) A Super Duper Jean company has 3 designs that can be made with short or long length. There are 5 color patterns available. How many different types of jeans are available from this company? A) 15 B) 30 C) 10 D) 25 E) 8

74)

75)

Use a Venn Diagram and the given information to determine the number of elements in the indicated region. 76) At Southern States University SSU) there are 719 students taking Finite Mathematics or 76) Statistics. 328 are taking Finite Mathematics, 476 are taking Statistics, and 85 are taking both Finite Mathematics and Statistics. How many are taking Statistics but not Finite Mathematics? A) 634 B) 391 C) 243 D) 158 Evaluate. 77) C7, 0 A) 1

77) B) 7

C) 6

D) 720

Provide an appropriate response. 78) License plates are made using 3 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed? A) 17,576,000 B) 308,915,776 C) 175,760 D) 1,757,600 E) 1,000,000

78)

Let A = {1, 3, 5, 7}; B = {5, 6, 7, 8}; C = {5, 8}; D = {2, 5, 8}; and U = {1, 2, 3, 4, 5, 6, 7, 8}. Determine whether the given statement is true or false. 79) A ⊂ A 79) A) True B) False Construct a truth table to decide if the two statements are equivalent. 80) ~p ∧ ~q; ~(p ∨ q) A) True B) False

80)

Solve the problem. 81) If the police have 7 suspects, how many different ways can they select 5 for a lineup? A) 21 B) 35 C) 2520 D) 42

81)

Use the Venn diagram to find the requested set. 82) Find A' ∩ B'.

82)

4 3

A) ∅

B) {7}

C) {7, 3, 4, z, r, h}

D) {4}

Construct a truth table to decide if the two statements are equivalent. 83) ~q ∧ p ; ~q → p A) True B) False

83)

Use the Venn diagram below to find the number of elements in the region.

84) n(A ∩ C) A) 2

84) B) 18

C) 10

D) 37

Construct a truth table for the proposition. 85) ~p ∧ ~q A) p q (~p ∧ ~q) T T T T F F F T F F F T C) p q (~p ∧ ~q) T T F T F T F T T F F T

85) B) p T T F F

q (~p ∧ ~q) T F F F T F F T

p T T F F

q (~p ∧ ~q) T F F F T F F F

Let A = {6, 4, 1, {3, 0, 8}, {9}}. Determine whether the statement is true or false. 86) {9} ∈ A A) True B) False

86)

Evaluate. 87) P 6, 4

87)

A) 2

B) 360

C) 30

D) 24

Use a Venn Diagram and the given information to determine the number of elements in the indicated region. 88) n(A) = 33, n(B) = 19, n(A ∩ B) = 1, n(A' ∩ B') = 9. Find n(U). 88) A) 51 B) 60 C) 64 D) 52 Evaluate. 89)

9! 7! 2! A) 9

89) B) 72

C) 1

D) 36

Use the Venn diagram below to find the number of elements in the region.

90) n(A) A) 17

90) B) 9

C) 12

D) 4

Provide an appropriate response. 91) A local television station sends out questionnaires to determine if viewers would rather see a documentary, an interview show, or reruns of a game show. There were 300 responses with the following results:

91)

90 were interested in an interview show and a documentary, but not reruns. 12 were interested in an interview show and reruns but not a documentary 42 were interested in reruns but not an interview show. 72 were interested in an interview show but not a documentary. 30 were interested in a documentary and reruns. 18 were interested in an interview show and reruns. 24 were interested in none of the three. How many are interested in exactly one kind of show? A) 154 B) 144 C) 124

D) 134

Evaluate. 92)

16! 15! A) 15

92) B) 32

C) 8

D) 16

Answer Key Testname: CHAP 07_14E

1) B 2) B 3) (6)(5)(5)(2) = 300 4) 6,720 5) p q p ∨ q ~(p ∨q) ~p ~q ~p ∧~q T T T F F F F T F T F F T F F T T F T F F F F F T T T T It is true since the fourth column and the seventh column are identical. 6) 9 9 = 387,420,489 7) {a, i, t, e} 8) 45,927 9) Converse (q → p): If n is an integer that is a multiple of 3 and a multiple of 5, then n is an integer that is a multiple of 15. Contrapositive (~q → ~p): If n is an integer that is not a multiple of 3 and 5, then n is not a multiple of 15. 10) p ~p q ~p ∨ q T F T T T F F F F T T T F T F T Contingency 11) 440 12) (44)(28 ) = 65,536 13) {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (H, 7), (T, H), (T, T)} 14) q p q ∨ p ~q (q ∨ p) ∧ ~q T T T T F T F T T F F F Contingency 15) B 16) D 17) B 18) B 19) B 20) A 21) A 22) C 23) A 24) B 25) D 26) B 27) B 28) A

F F T T

F F T F

Answer Key Testname: CHAP 07_14E

29) D 30) B 31) C 32) D 33) A 34) A 35) D 36) A 37) A 38) B 39) A 40) A 41) A 42) D 43) D 44) A 45) A 46) A 47) A 48) C 49) C 50) A 51) C 52) D 53) A 54) D 55) D 56) B 57) D 58) D 59) A 60) D 61) C 62) D 63) A 64) B 65) C 66) C 67) B 68) D 69) B 70) B 71) A 72) D 73) A 74) A 75) B 76) B 77) A 15

Answer Key Testname: CHAP 07_14E

78) A 79) B 80) A 81) A 82) A 83) B 84) C 85) B 86) A 87) B 88) B 89) D 90) A 91) B 92) D

CHAPTER 8

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the probability. 1) Refer to the table below for events in a sample space, S, compute P(C|E).

D E Totals

A 0.12 0.03 0.15

B 0.50 0.10 0.60

C 0.08 0.17 0.25

Total 0.70 0.30 1.00

Provide an appropriate response. 2) The payoff table for three possible courses of action A1, A2, and A3 is given below.

pi .3 .2 .1 .4

xi $70 $100 $160 $80

xi $40 $120 $140 $140

xi $50 $110 $90 $160

Which course of action will produce the largest expected value? What is it? List the outcomes of the sample space. 3) A fair die and a fair coin are tossed in succession. Find the sample space composed of equally likely events. Find the probability. 4) A class of 40 students has 10 honor students and 13 athletes. Three of the honor students are also athletes. One student is chosen at random. Find the probability that this student is an athlete if it is known that the student is not an honor student. Solve the problem. 5) Two groups of people were asked their preference in television programs from among three new programs. The results are shown in the table below. What is the probability that a person selected at random will be from group A or prefer program X?

Group of People A B

X 25 20

Television Program Y 15 60

Z 40 50

Provide an appropriate response. 6) A shipment of 20 digital cameras contains two that are defective. A random sample of three is selected and tested. Let X be the random variable associated with the number of defective cameras in a sample. Find the probability distribution of X and the expected number of defective cameras in a sample. Find the probability. 7) One urn has 4 red balls and 1 white ball; a second urn has 2 red balls and 3 white balls. A single card is randomly selected from a standard deck. If the card is less than 5 (aces count as 1), a ball is drawn out of the first urn; otherwise a ball is drawn out of the second urn. If the drawn ball is red, what is the probability that it came out of the second urn?

8) A basketball team is to play two games in a tournament. The probability of winning the first game is .10. If the first game is won, the probability of winning the second game is .15. If the first game is lost, the probability of winning the second game is .25. What is the probability the first game was won if the second game is lost?

9) Each person in a group of students was identified by his or her hair color and then asked whether he or she preferred taking classes in the morning, afternoon, or evening. The results are shown in the table below. Find the probability that a student preferred morning classes given that he or she has blonde hair. Hair Color Class Time Preference Blonde Brunette Rehead Morning 45 25 10 Afternoon 40 15 50 Evening 35 20 30

Solve the problem. 10) From a survey involving 2,000 students at a large university, it was found that 1,300 students had classes on Monday, Wednesday, and Friday; 1,500 students had classes on Tuesday and Thursday; and 800 students had classes every day. If a student at this university is selected at random, what is the (empirical) probability that the student has classes only on Tuesday and Thursday? Find the probability. 11) A box contains 7 red balls and 3 white balls. Two balls are to be drawn in succession without replacement. What is the probability that the sample will contain exactly one white ball and one red ball?

10)

11)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use Bayes' rule to find the indicated probability. 12) Two stores sell a certain MP3 players. Store A has 34% of the sales, 5% of which are of defective items, and store B has 66% of the sales, 1% of which are of defective items. The difference in defective rates is due to different levels of pre -sale checking of the product. A person receives a defective item of this product as a gift. What is the probability it came from store B? A) 0.22 B) 0.275 C) 0.5667 D) 0.7083

12)

Find the probability. 13) Two 6 -sided dice are rolled. What is the probability that the sum of the two numbers on the dice will be greater than 9? 1 1 1 A) B) C) 6 D) 6 4 12 14) The table below describes the smoking habits of a group of asthma sufferers.

Men Women Total

Nonsmoker 363 443 806

Occasional Regular smoker smoker 33 89 40 66 73 155

13)

14)

Heavy smoker Total 46 531 44 593 90 1124

If one of the 1124 people is randomly selected, find the probability that the person is a man or a heavy smoker. A) 0.512 B) 0.511 C) 0.471 D) 0.552 Estimate the indicated probability. 15) College students were given three choices of pizza toppings and asked to choose one favorite. The following table shows the results. toppings freshman sophomore cheese 12 12 meat 27 24 veggie 12 12

junior 23 12 27

15)

senior 24 12 24

A randomly selected student prefers a cheese topping. A) .321 B) .109 C) .338

D) .339

Use Bayes' rule to find the indicated probability. 16) An water well is to be drilled in the desert where the soil is either rock, clay or sand. The probability of rock P(R) = 0.53. The clay probability is P(C) = 0.21. The sand probability is P(S) = 0.26. It if it rock, a geological test gives a positive result with 35% accuracy. If it is clay, this test gives a positive result with 48% accuracy. The test gives a 75% accuracy for sand. Given the test is positive, what is the probability that soil is rock, P(sand | positive)? A) P(sand | positive) = 0.209 B) P(sand | positive) = 0.385 C) P(sand | positive) = 0.405 D) P(sand | positive) = 0.26

16)

Find the probability. 17) People were given three choices of soft drinks and asked to choose one favorite. The following table shows the results.

under 18 years of age

17)

diet cola root beer lemon-drop 40 25 20

between 18 and 40 over 40 years of age

35 20

20 30

30 35

Find P(person is over 40|person drinks root beer). 6 2 A) B) 17 5

20 17

5 17

18) A sample of 4 different calculators is randomly selected from a group containing 14 that are defective and 26 that have no defects. What is the probability that at least one of the calculators is defective? A) 0.821 B) 0.164 C) 0.140 D) 0.836

18)

19) The table lists the eight possible blood types. Also given is the percent of the U.S. population having that type.

19)

Type/RH factor Percent O positive 38% A positive 34% B Positive 9% O Negative 7% A Negative 6% AB Positive 3% B Negative 2% AB Negative 1% Let E be the event that a randomly selected person in the U. S. has type B blood. Let F be the event that a randomly selected person in the U. S. has RH-positive blood. Find P(E|F). A) 0.840 B) 0.182 C) 0.107 D) 0.818 Find the expected value. 20) A car agency has found daily demand to be as shown in the table. Number of customers 7 8 9 10 11 Probability 0.10 0.20 0.40 0.20 0.10 Find the expected number of customers. A) 2 B) 12

C) 10

20)

D) 9

Use Bayes' rule to find the indicated probability. 21) An water well is to be drilled in the desert where the soil is either rock, clay or sand. The probability of rock P(R) = 0.53. The clay probability is P(C) = 0.21. The sand probability is P(S) = 0.26. It if it rock, a geological test gives a positive result with 35% accuracy. If it is clay, this test gives a positive result with 48% accuracy. The test gives a 75% accuracy for sand. Given the test is positive, what is the probability that soil is rock, P(rock | positive)? A) P(rock | positive) = 0.405 B) P(rock | positive) = 0.53 C) P(rock | positive) = 0.209 D) P(rock | positive) = 0.385 Find the probability. 22) A bag contains 6 red marbles, 2 blue marbles, and 1 green marble. What is the probability of choosing a marble that is not blue? 7 9 2 A) B) 7 C) D) 9 7 9 Determine independence. Answer Yes or No. 23) According to a survey, 8% of students at a college are left handed, 53% are female, and 4.24% are both female and left handed. Is being left handed independent of gender? A) Yes B) No Solve the problem. 24) The distribution of bachelor degrees conferred by a local college is listed below, by major. Major English Mathematics Chemistry Physics Liberal Arts Business Engineering

21)

22)

23)

24)

Frequency 2073 2164 318 856 1358 1676 868 9313

What is the probability that a randomly selected degree is not in Mathematics? A) 0.768 B) 0.303 C) 0.682 D) 0.232 Use Bayes' rule to find the indicated probability. 25) Quality Motors has three plants. Plant 1 produces 35% of the car output, plant 2 produces 20% and plant 3 produces the remaining 45%. One percent of the output of plant 1 is defective, 1.8% of the output of plant 2 is defective and 2% of the output of plant 3 is defective. The annual total production of Quality Motors is 1,000,000 cars. A car chosen at random from the annual output and is found defection. What is the probability that it came from plant 2? A) 0.559 B) 0.217 C) 0.35 D) 0.224 List the outcomes of the sample space. 26) A box contains 10 blue cards numbered 1 through 10. List the sample space of picking one card from the box. A) {8} B) {100} C) {10} D) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

25)

26)

Provide an appropriate response. 27) In a raffle 1000 tickets are being sold at $1.00 each. The first prize is $100, and there are 3 second prizes fo $50 each. By how much does the price of a ticket exceed its expected value? A) $0.75 B) $1.75 C) $1.00 D) $50

27)

The graduates at a southern university are shown in the table.

Male, M Female, F Total

Art & Science A 342 324 666

Education E 424 102 526

Business B 682 144 826

Total 1448 570 2018

A student is selected at random from the graduating class. 28) Find the probability that the student is female, given that the student is receiving an education degree, P(F|E). A) P(F|E) = 570 B) P(F|E) = 51 2018 263 C) P(F|E) = 324 666

D) P(F|E) = 724 1009

List the outcomes of the sample space. 29) A box contains 10 red cards numbered 1 through 10. List the sample space of picking one card from the box. A) {10} B) {1, 10} C) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} D) {100} Find the probability. 30) A bag contains 5 red marbles, 2 blue marbles, and 4 green marbles. What is the probability of choosing a blue marble? 5 4 2 2 A) P(B) = B) P(B) = C) P(B) = D) P(B) = 11 11 7 11 31) A calculator requires a keystroke assembly and a logic circuit. Assume that 98% of the keystroke assemblies and 93% of the logic circuits are satisfactory. Find the probability that a finished calculator will be satisfactory. Assume that defects in keystroke assemblies are independent of defects in logic circuits. A) .9600 B) .8649 C) .9604 D) .9114 Provide an appropriate response. 32) The probability distribution for the random variable X is: xi -1 0 1 2 p i 0.22

0.23

28)

0.26

C) 0.22

30)

31)

32)

0.29

What is the expected value of X? A) 0.50 B) 0.26

29)

D) 0.62

Find the indicated probability. 33) In a homicide case 6 different witnesses picked the same man from a line up. The line up contained 5 men. If the identifications were made by random guesses, find the probability that all 6 witnesses would pick the same person. A) 0.00032 B) 0.000064 C) 0.0001286 D) 1.2 Determine independence. Answer Yes or No. 34) A group of 25 people contains 10 brunettes, 8 blondes, and 7 redheads. Of the 20 girls in the group, 8 are brunettes, 6 are blondes, and 6 are redheads. A person is selected at random. Are the events of being a girl and having brown hair independent? A) Yes B) No Solve the problem. 35) At the Stop 'n Go tune -up and brake shop, the manager has found that an SUV will require a tune -up with a probability of 0.6, a brake job with a probability of 0.1 and both with a probability of 0.02. What is the probability that an SUV requires neither type of repair? A) 0.58 B) 0.68 C) 0.7 D) 0.32 Determine independence. Answer Yes or No. 36) The table shows the political affiliation of voters in one city and their positions on stronger gun control laws.

33)

34)

35)

36)

Stronger Gun Control Favor Oppose Republican 0.08 0.33 Democrat 0.22 0.20 Other 0.13 0.04 Are party affiliation and position on gun control laws independent? A) Yes B) No Find the probability. 37) A sample space consists of 118 separate events that are equally likely. What is the probability of each? 1 A) 1 B) 0 C) D) 118 118

37)

The graduates at a southern university are shown in the table.

Male, M Female, F Total

Art & Science A 342 324 666

Education E 424 102 526

Business B 682 144 826

Total 1448 570 2018

A student is selected at random from the graduating class. 38) Find the probability that the student is female, given that an education degree is not received, P(F|E'). A) P(F|E') = 117 B) P(F|E') = 324 373 666 C) P(F|E') = 424 526

38)

D) P(F|E') = 102 526

Find the probability. 39) A lottery game contains 28 balls numbered 1 through 28. What is the probability of choosing a ball numbered 28, P(28)? A) P(28) = 1 B) P(28) = 0 C) P(28) = 28 D) P(28) = 1 28 Use the tree diagram to find the requested probability. 40) Find P(M∣Q). Give your answer as a fraction.

39)

40)

a = 0.8 , b = 0.2, c = 0.6 , d = 0.2, e = 0.2, f = 0.5 , g = 0.4, h = 0.1 2 1 2 A) B) C) 25 3 5

6 25

Find the probability. 41) A lottery game has balls numbered 1 through 21. A randomly selected ball has an even number or a 10. 10 7 2 A) B) C) 10 D) 21 2 7 Use a tree diagram to find the indicated probability. 42) 3.9% of a secluded island tribe are infected with a certain disease. There is a test for the disease, however the test is not completely accurate. 92% of those who have the disease will test positive. However 4.4% of those who do not have the disease will also test positive (false positives). What is the probability that any given person will test positive? Round your answer to three decimal places if necessary. A) 0.078 B) 0.036 C) 0.482 D) 0.042

41)

42)

Find the indicated probability. 43) In a family with family with 4 children, excluding multiple births, what is the probability of having 2 girls and 2 boys, in that order? Assume that a boy is as likely as a girl at each birth. 1 1 1 1 A) B) C) D) 16 8 2 4

43)

In a survey of the number of DVDs in a house, the table shows the probabilities. Number of DVDs 0 1 2 3 4 or more Probability 0.05 0.024 0.33 0.21 0.17 44) Find the probability of a house having fewer than 2 DVDs. A) 0.57 B) 0.29 C) 0.38

44) D) 0.71

List the outcomes of the sample space. 45) A 6-sided die is rolled. The sides contain the numbers 1, 2, 3, 4, 5, 6. List the sample space of rolling one die. A) {1, 2, 3, 4, 5, 6} B) {1, 2, 6} C) {36} D) {6} Use Bayes' rule to find the indicated probability. 46) In Cumberland County, 55% of registered voters are Democrats, 30% are Republicans and 15% are independent. During a recent election, 35% of the Democrats voted, 65% of the Republicans voted, and 75% of the independent voted. What is the probability that someone who voted is a Republican? A) 0.385 B) 0.225 C) 0.39 D) 0.065

45)

46)

In a survey of the number of DVDs in a house, the table shows the probabilities. Number of DVDs 0 1 2 3 4 or more Probability 0.05 0.024 0.33 0.21 0.17 47) Find the probability of a house having 1 or 2 DVDs. A) 0.38 B) 0.83 C) 0.57

47) D) 0.95

Find the odds. 48) Suppose you are playing a game of chance. If you bet $9 on a certain event, you will collect $360(including your $9 bet) if you win. Find the odds used for determining the payoff. A) 360:369 B) 40:1 C) 39:1 D) 1:39 Find the expected value. 49) A new light bulb has been found to have a 0.02 probability of being defective. A shop owner receives 500 bulbs of this kind. How many of these bulbs are expected to be defective? A) 100 B) 500 C) 20 D) 10

48)

49)

The graduates at a southern university are shown in the table.

Male, M Female, F Total

Art & Science A 342 324 666

Education E 424 102 526

Business B 682 144 826

Total 1448 570 2018

A student is selected at random from the graduating class. 50) Find the probability that the student is male, P(M). A) P(M) = 724 B) P(M) = 424 1009 526

50) C) P(M) = 285 1009

D) P(M) = 171 724

Estimate the indicated probability. 51) College students were given three choices of pizza toppings and asked to choose one favorite. The following table shows the results. toppings freshman sophomore cheese 11 11 meat 29 19 veggie 11 11

junior 23 11 29

51)

senior 19 11 19

A randomly selected student prefers a cheese topping. A) .0 54 B) .172 C) .314

D) .569

Find the probability. 52) Find the probability of correctly answering the first 5 questions on a multiple choice test if random guesses are made and each question has 4 possible answers. 1 4 5 1 A) B) C) D) 1024 5 4 625 53) A packet of sour worms contains four strawberry, four lime, two black currant, two orange sour, and three green apples worms. What is the probability that Dustin will choose a green apple sour worm, P(green apple)? A) P(green apple) = 0 B) P(green apple) = 1 5 C) P(green apple) = 3 5

52)

53)

D) P(green apple) = 1 15

Prepare a probability distribution for the experiment. Let x represent the random variable, and let P represent the probability. 54) Two marbles are drawn from a bag in which there are 4 red marble and 2 blue marble. The 54) number of blue marbles is counted. A) B) C) D) x P x P x P x P 0 0.333 0 0.07 0 0.4 0 0.719 1 0.333 1 0.53 1 0.53 1 0.280 2 0.333 2 0.4 2 0.07 2 0.001

Find the probability. 55) People were given three choices of soft drinks and asked to choose one favorite. The following table shows the results.

under 18 years of age

55)

diet cola root beer lemon drop 40 25 20

between 18 and 40 over 40 years of age

35 20

20 30

30 35

P(person is over 40 ∩ person drinks diet cola)? 4 4 A) B) 51 19

30 19

4 17

Solve the problem. 56) At the Stop 'n Go tune -up and brake shop, the manager has found that an SUV will require a tune -up with a probability of 0.6, a brake job with a probability of 0.1 and both with a probability of 0.02. What is the probability that an SUV requires a tune-up but not a brake job? A) 0.7 B) 0.02 C) 0.68 D) 0.58

56)

Prepare a probability distribution for the experiment. Let x represent the random variable, and let P represent the probability. 57) Three coins are tossed, and the number of heads is noted. 57) A) B) C) D) x P x P x P x P 0 1/8 0 3/16 0 1/6 0 1/3 1 3/8 1 5/16 1 1/3 1 1/6 2 3/8 2 5/16 2 1/3 2 1/6 3 1/8 3 3/16 3 1/6 3 1/3 Find the probability. 58) The table lists the eight possible blood types. Also given is the percent of the U.S. population having that type. Type/RH factor Percent O positive 38% A positive 34% B Positive 9% O Negative 7% A Negative 6% AB Positive 3% B Negative 2% AB Negative 1% Let E be the event that a randomly selected person in the U. S. has type B blood. Let F be the event that a randomly selected person in the U. S. has RH-positive blood. Find P(F|E). A) 0.720 B) 0.818 C) 0.107 D) 0.340

58)

Use a tree diagram to find the indicated probability. 59) In the town of Cheraw, a certain type of laptop computer is sold at just two stores. Store A has 38% of the sales, 4% of which are of defective items, and store B has 62% of the sales, 2% of which are of defective items. A person receives one of these laptop computers as a gift. What is the probability it is defective? A) 0.028 B) 0.014 C) 0.03 D) 0.42 Find the odds. 60) If two fair dice are thrown, what are the odds of obtaining a sum of 7? A) 3:4 B) 1:6 C) 1:5

60) D) 5:1

Use the tree diagram to find the requested probability. 61) Find P(X∣A). Give your answer as a decimal and round your answer to three decimal places if necessary.

a = 0.9 , b = 0.1, c = 0.6 , d = 0.4, e = 0.2, f = 0.8 A) 0.54 B) 0.675

C) 0.964

59)

61)

D) 0.6

Find the probability. 62) Two 6 -sided dice are rolled. What is the probability the sum of the two numbers on the die will be 5, P(sum of 5)? A) P(sum of 5) = 5 B) P(sum of 5) = 1 6 9

62)

D) P(sum of 5) = 8 9

C) P(sum of 5) = 4

Find the expected value. 63) Mr. Cameron is sponsoring an summer concert. He estimates that he will make $300,000 if it does not rain and make $60,000 if it does rain. The weather bureau predicts the chance of rain is 0.34 for the day of the concert. An insurance company is willing to insure the concert for $150,000 against rain for a premium of $30,000. If he buys this policy, what are his expected earnings from the concert? A) $300,000 B) $270,000 C) $180,000 D) $239,400 List the outcomes of the sample space. 64) There are 3 balls in a hat; one with the number 2 on it, one with the number 6 on it, and one with the number 8 on it. You pick a ball from the hat at random and then you flip a coin. Using a tree diagram , obtain the sample space for the experiment. List the elements that make up the sample space. A) 2 H, 2 T, 6 H, 6 T, 8 H, 8 T B) 2 6 8 H, 2 6 8 T C) 2 H, 6 H, 8 H D) 2 H T, 6 H T, 8 H T

63)

64)

Use the tree diagram to find the requested probability. 65) Find P(X∣A). Give your answer as a decimal and round your answer to three decimal places if necessary.

a = 0.9 , b = 0.1, c = 0.8 , d = 0.2, e = 0.6 , f = 0.4 A) 0.514 B) 0.8

C) 0.923

D) 0.72

Find the expected value. 66) Suppose that 1,000 tickets are sold for a raffle that has the following prizes: one $300 prize, two $100 prizes, and one hundred $1 prizes. What is expected value of a ticket? A) $100 B) $0.60 C) $1 D) $300 Provide an appropriate response. 67) The number of loaves of whole wheat bread left on the shelf of a local quick stop at closing (denoted by the random variable X) varies from day to day. Past records show that the probability distribution of X is as shown in the following table. Find the probability that there will be at least three loaves left over at the end of any given day. xi 0 p i 0.20 A) 0.20

0.25 0.20

0.15

0.10

0.08

0.02

B) 0.35

C) 0.65

66)

67)

D) 0.15

Use Bayes' rule to find the indicated probability. 68) In Cumberland County, 55% of registered voters are Democrats, 30% are Republicans and 15% are independent. During a recent election, 35% of the Democrats voted, 65% of the Republicans voted, and 75% of the independent voted. What is the probability that someone who voted is a Democrat? A) 0.615 B) 0.45 C) 0.55 D) 0.385 Find the probability. 69) Samantha is taking courses in math and English. The probability of passing math is estimated at 0.4 and English at 0.6. She also estimates that the probability of passing at least one of them is 0.8. What is her probability of passing both courses? A) 0.8 B) 0.2 C) 0 D) 0.12 70) In a batch of 8,000 clock radios 6% are defective. A sample of 11 clock radios is randomly selected without replacement from the 8,000 and tested. The entire batch will be rejected if at least one of those tested is defective. What is the probability that the entire batch will be rejected? A) 0.0600 B) 0.506 C) 0.0909 D) 0.494

65)

68)

69)

70)

71) The table shows the political affiliation of voters in a small midwestern town and their positions on stronger drug control laws.

71)

Stronger Drug Control Favor Oppose Republican 0.11 0.27 Democrat 0.25 0.16 Other 0.15 0.06 Find the probability that a Democrat opposes stronger drug control laws. A) 0.490 B) 0.420 C) 0.390

D) 0.350

List the outcomes of the sample space. 72) A group of 18 songs are assigned numbers 1 through 18. List the outcomes of sample space of the event choosing a song with a number 5 or less. A) {1, 2, 3, 4, 5} B) {18} C) {1, 2, 3, 4} D) {1} Find the probability. 73) A packet of sour worms contains four strawberry, four lime, two black currant, two orange sour, and three green apples worms. What is the probability that Dylan will not choose a green apple sour worm, P(not green apple)? A) P(not green apple) = 4 B) P(not green apple) = 3 15 15 C) P(not green apple) = 4 5

72)

73)

D) P(not green apple) = 0

Use Bayes' rule to find the indicated probability. 74) Two shipments of components were received by a factory and stored in two separate bins. Shipment I has 2% of its contents defective, while shipment II has 5% of its contents defective. If it is equally likely an employee will go to either bin and select a component randomly, what is the probability that a defective component came from shipment II? A) 0.714 B) 0.2 C) 0.222 D) 0.5 Find the probability. 75) A single fair die is rolled. The number on the die is a 3 or a 5. 1 1 A) 2 B) C) 6 36

74)

75) D)

1 3

76) If 80% of scheduled flights actually take place and cancellations are independent events, what is the probability that 3 separate flights will all take place? A) .64 B) .0 1 C) .80 D) .51

76)

In a survey of the number of DVDs in a house, the table shows the probabilities. Number of DVDs 0 1 2 3 4 or more Probability 0.05 0.024 0.33 0.21 0.17 77) Find the probability of a house having 3 or more DVDs. A) 0.57 B) 0.05 C) 0.83

77) D) 0.38

Find the odds. 78) If the sectors are of equal size, what are the odds of spinning an A on this spinner?

A) 4:2

B) 2:6

C) 3:5

D) 6:2

Find the probability. 79) A bag contains 15 balls numbered 1 through 15. What is the probability of selecting a ball that has an even number? 7 15 2 A) B) C) 7 D) 15 7 15 Find the odds. 80) In a certain town, 20% of people commute to work by bus. If a person is selected randomly from the town, what are the odds against selecting someone who commutes by bus? A) 4:5 B) 4:1 C) 1:4 D) 1:5 Find the probability. 81) A 6-sided die is rolled. What is the probability of rolling a number less than 2? 1 1 5 1 A) B) C) D) 3 3 6 6 82) A study conducted at a certain college shows that 62% of the school's graduates find a job in their chosen field within a year after graduation. Find the probability that among 5 randomly selected graduates, at least one finds a job in his or her chosen field within a year of graduating. A) 0.992 B) 0.200 C) 0.620 D) 0.908 Solve the problem. 83) A drug company is running trials on a new test for anabolic steroids. The company uses the test on 400 athletes know to be suing steroids and 200 athletes known not to be using steroids. Of those using steroids, the new test is positive for 390 and negative for 10. Of those not using steroids, the test is positive for 10 and negative for 190. What is the estimated probability of a false negative result (the probability that an athlete using steroids will test negative)? A) 0.05 B) 0.025 C) 0.975 D) 0.95

78)

79)

80)

81)

82)

83)

Estimate the indicated probability. 84) College students were given three choices of pizza toppings and asked to choose one favorite. The following table shows the results. toppings freshman sophomore

junior

senior

cheese

meat

veggie

84)

A randomly selected student who is a a junior or senior is also prefers veggie. Round the answer to the nearest hundredth. A) .638 B) .386 C) .212 D) .375 Find the probability. 85) At the Stop 'n Go tune -up and brake shop, the manager has found that an SUV will require a tune -up with a probability of 0.6, a brake job with a probability of 0.1 and both with a probability of 0.02. What is the probability that an SUV requires either a tune-up or a brake job? A) 0.7 B) 0.58 C) 0.32 D) 0.68 Find the expected value. 86) A fair coin is tossed three times, and a player wins $3 if 3 tails occur, wins $2 if 2 tails occur and loses $3 if no tails occur. If one tail occurs, no one wins. What is the expected value of the games? A) $3.00 B) $2.00 C) -$3.00 D) $0.75 Solve the problem. 87) Of the coffee makers sold in an appliance store, 4.0% have either a faulty switch or a defective cord, 2.5% have a faulty switch, and 0.1% have both defects. What is the probability that a coffee maker will have a defective cord? Express the answer as a percentage. A) 2.6% B) 4.0% C) 1.6% D) 4.1% Find the odds. 88) The probability of a person getting a job interview is 0.54. What are the odds against getting the interview? A) 54:100 B) 23:54 C) 1:54 D) 23:27 Find the expected value. 89) Mr. Cameron is sponsoring an summer concert. He estimates that he will make $300,000 if it does not rain and make $60,000 if it does rain. The weather bureau predicts the chance of rain is 0.34 for the day of the concert. What are Mr. Cameron's expected concert earning? A) $60,000 B) $300,000 C) $218,400 D) $360,000

85)

86)

87)

88)

89)

Use Bayes' rule to find the indicated probability. 90) An water well is to be drilled in the desert where the soil is either rock, clay or sand. The probability of rock P(R) = 0.53. The clay probability is P(C) = 0.21. The sand probability is P(S) = 0.26. It if it rock, a geological test gives a positive result with 35% accuracy. If it is clay, this test gives a positive result with 48% accuracy. The test gives a 75% accuracy for sand. Given the test is positive, what is the probability that soil is clay, P(clay | positive)? A) P(clay | positive) = 0.385 B) P(clay | positive) = 0.53 C) P(clay | positive) = 0.405 D) P(clay | positive) = 0.209 91) The incidence of a certain disease on the island of Tukow is 4%. A new test has been developed to diagnose the disease. Using this test, 91% of those who have the disease test positive while 4% of those who do not have the disease test positive (false positive). If a person tests positive, what is the probability that he or she actually has the disease? A) 0.487 B) 0.91 C) 0.856 D) 0.438

90)

91)

Answer Key Testname: CHAP 08_14E

1) 0.5667 2) A3 has the largest expected value. E(A3 ) = 110 3) {(1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T)} 1 ≈ 0.333 4) 3 5)

100 ≈ 0.476 210

6) xi 0 1 2 ; 0.3 p i 0.7158 0.2684 0.0158 7) P(Urn 2 R) = 9 = 0.5294 17 8) P( win first game | lose second game) = 17 ≈ 0.1118 152 9)

45 = 0.375 120

10) 0.350 42 ≈ 0.467 11) 90 12) B 13) A 14) A 15) A 16) C 17) B 18) D 19) D 20) D 21) D 22) A 23) A 24) A 25) D 26) D 27) A 28) B 29) C 30) D 31) D 32) D 33) B 34) A 35) D 36) B 37) C 38) A 18

Answer Key Testname: CHAP 08_14E

39) A 40) B 41) A 42) A 43) A 44) B 45) A 46) C 47) C 48) C 49) D 50) A 51) C 52) A 53) B 54) C 55) A 56) D 57) A 58) C 59) A 60) C 61) C 62) B 63) D 64) A 65) C 66) B 67) B 68) D 69) B 70) D 71) C 72) A 73) C 74) A 75) D 76) D 77) D 78) C 79) A 80) B 81) D 82) A 83) B 84) B 85) D 86) D 87) C 19

Answer Key Testname: CHAP 08_14E

88) D 89) C 90) D 91) A

CHAPTER 9

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 0.1 0.1 0.8 1) Find the stationary matrix for the transition matrix P = 0.3 0.3 0.4 0.4 0.4 0.2 Round the numbers in your answer to the nearest hundredth. A) [0.29 0.29 0.42] B) [0.26 0.44 0.30] C) [0.23 0.17 0.60] D) [0.17 0.29 0.54]

2) Decide whether or not the transition matrix is regular. Answer Yes or No. 1 0 0.5 0.5 A) Yes B) No

3) The probability that a car owner will become a car renter in five years is 0.03. The probability that a renter will become an owner in five years is 0.1. Suppose the proportions in the population are 64% owners (O), 35.5% renters (R) and .5% neither (N) with the following transition matrix.

O R N O 0.94 0.06 0 R 0.12 0.879 0.001 N 0 32 0.68 Find the long-range probabilities for the three categories. A) [0.94 0.879 0.001] B) [0.666 0.333 0.001] C) [0.64 0.355 0.005] D) [0.94 0.06 0] 4) Find the limiting matrix P corresponding to the transition matrix P = 0.8 0.2 . 0.1 0.9 Round to the nearest hundredths. A) 0.66 0.34 B) [0.875 0.125] C) 0.33 0.67 D) 0.45 0.55 0.17 0.83 0.33 0.67 0.28 0.72

5) A trailer rental company has rental and return facilities at both a north and south location in a city. Assume a trailer must be returned to one or the other of these locations. If a trailer is rented at the north location, the probability that it will be returned there is .6; if a trailer is rented at the south location, the probability it will be returned there is .65. Assume the company rents all of its trailers each day and each trailer is rented (and returned) only once a day. If the company starts with 50% of the trailers at each location, what is the expected distribution (in percentages) the next day? A) 37.5% of the trailers at the north location; 62.5% of the trailers at the south location B) 62.5% of the trailers at the north location; 37.5% of the trailers at the south location C) 47.5% of the trailers at the north location; 52.5% of the trailers at the south location D) 52.5% of the trailers at the north location; 47.5% of the trailers at the south location

6) According to data collected during one year in a large metropolitan community, 30% of commuters used public transportation to get to work, and this rose by 4% the following year. This is modeled by the transition matrix

′

P P P 0.9 0.1 M= ′ P 0.1 0.9 ′

where P represents the percentage of people that use public transportation and P the percentage of people that do not. Let S 0 = [0.3 0.7]. Find S 2. Round your answer to the nearest thousandths. A) [0.34 0.66] B) [0.628 0.372] 7) Given the transition matrix: A B C D A 0.4 0.3 0.2 0.1 1 0 0 P= B 0 C 0 0 1 0 D 0.2 0.5 0.1 0.2

C) [0.372 0.628]

D) [0.66 0.34] 7)

P4 =

A B C D

A B C D 0.0396 0.5942 0.3518 0.0144 0 1 0 0 0 0 1 0 0.0288 0.756 0.2044 0.0108

Find the probability of going from state C to state B in four trials. A) 0.756 B) 0.0396 C) 0.0144

D) 0

8) A red urn contains 4 red marbles, 2 blue marbles, and 4 green marbles. A blue urn contains 2 red marbles, 2 blue marbles, and 1 green marble. A green urn contains 3 green marbles. A marble is selected from an urn, the color is noted, and the marble is returned to the urn from which it was drawn. The next marble is drawn from the urn whose color is the same as the marble just drawn. Thus, this is a Markov process with three states: draw from the red urn, draw from the blue urn, or draw from the green urn. Write the transition matrix P. A) G B R B) G B R G 3 0 0 G 1 0 0 P= B 2 2 1 P= B 0 1 0 R 4 2 4 R 0 0 1 C) G B R D) G B R G 1 0 0 G 0.4 0.2 0.4 P = B 0.2 0.4 0.4 P = B 0.2 0.4 0.4 R 0.4 0.2 0.4 R 0.4 0.2 0.4

A small town has only two dry cleaners, Fast and Speedy. Fast hopes to increase its market share by conducting an extensive advertising campaign. The initial market share for Fast was 40% and 60% for Speedy. Solve the problem. 9) Find the probability that a customer using Fast initially will use Fast for his second batch of 9) clothes. Use the following transition matrix. Fast Speedy Fast

0.29

0.71

Speedy 0.73

0.27

A) 0.29

B) 0.27

C) 0.73

D) 0.71

Construct the transition matrix that represents the data. 10) A new anti-gravity commuter train has been installed. It is expected that each week 90% of the riders who used the existing system will continue to do so. Of those who traveled by car, 5% will begin to use the new anti-gravity train. Use this information to write the transition matrix that describes this process. T C T C A) T C B) T C T 0.95 0.10 T 0.90 0.05 C 0.05 0.95 C 0.10 0.95 C) T C D) T C T 0.90 0.10 T 0.95 0.10 C 0.05 0.95 C 0.05 0.90 Provide an appropriate response. 11) Decide whether or not the transition matrix is regular. Answer Yes or No. 0.7 0.3 0 0 0.4 0.6 1 0 0 A) Yes B) No 0.1 0.1 0.8 12) Find the limiting matrix P corresponding to the transition matrix P = 0.3 0.3 0.4 . 0.4 0.4 0.2 Round to the nearest hundredths. 0.1 0.1 0.8 0.29 0.29 0.29 A) 0.3 0.3 0.4 B) 0.29 0.29 0.29 0.4 0.4 0.2 0.29 0.29 0.29 0.36 0.36 0.28 1 0 0 C) 0.28 0.28 0.44 D) 0 1 0 0.24 0.24 0.52 0 0 1 13) Suppose that for a certain absorbing Markov chain the fundamental matrix is found to be $1 $2 $3 $1 1.5 1.0 0.5 $2 1.0 2.0 1.0 $3 0.5 1.0 1.5 What is the expected number of times a person will have $3, given that she started with $1? A) 1.0 B) 2.0 C) 0.5 D) 1.5

10)

11)

12)

13)

14) Find a standard form for the absorbing Markov chain with the transition matrix A B C 1 0 0 A 1 1 1 B 3 3 3 C 0 0 1 A) A C B B) A B C A 1 0 0 1 0 0 A C 0 1 0 0 1 0 B B 3 3 3 1 1 1 C 3 3 3 C)

A C B 1 0 0 0 1 0 1 1 1 3 3 3

A C B

D) A B C

14)

A B C 1 0 0 3 3 3 0 0 1

0.1 0.9 0 15) Find the stationary matrix for the transition matrix P = 0.3 0.4 0.3 . 0.2 0.2 0.6 2 4 1 9 9 9 A) 9 9 3 B) 2 4 3 1 0 0 C) 0 1 0 0 0 1

15)

D) [0.25 0.45 0.35]

16) Find the fundamental matrix F for the absorbing Markov chain with the given matrix. Express your answer in fraction form. 1 0 0 0 1 1 0 0 2 2 0 1 4 A)

0 1 2

1 0

3 2 1 1 2

0 1 4 B)

3 1 2 2

2 3 2 1

3 2

1 2

16)

17) The transition matrix for a Markov process is: State A B A 0.3 0.7 = P State B 0.9 0.1

17)

Find the first state matrix if the initial state is S 0 = .3 .7 . A) [0.468 0.532] B) [0.9 0.1] C) [0.3 0.7]

D) 0.72 0.28

0.2 0.6 0.2 18) Find the stationary matrix for the transition matrix P = 0.1 0.1 0.8 . 0.3 0.3 0.4 11 5 7 1 0 0 A) 23 23 23 B) 0 1 0 0 0 1 23 23 23 C) 5 D) [5 7 11] 11 7

18)

19) Decide whether or not the transition matrix is regular. Answer Yes or No. 0.3 0.7 0.9 0.1 A) Yes B) No

19)

20) From statistics gathered over many seasons, it was determined that the probability a basketball player will make a basket after having made a basket on his previous attempt is .55, while the probability he will make a basket if he missed on his previous attempt is .48. In a current game a player has made 45% of his attempted shots. If the player shoots many more times in the game, what would be the overall percentage of baskets that he makes in this game? A) 51% B) 48% C) 52% D) 49%

20)

21) According to data collected during one year in a large metropolitan community, 30% of commuters used public transportation to get to work, and this rose by 4% the following year. This is modeled by the transition matrix

21)

′

P P M = P ′ 0.9 0.1 P 0.1 0.9 ′

where P represents the percentage of people that use public transportation and P the percentage of people that do not. Let S 0 = [0.3 0.7]. Find S 1. A) [0.34 0.66]

B) [0.9 0.1]

C) [0.3 0.7]

22) Find the stationary matrix for the transition matrix P = 0.8 0.2 . 0.35 0.65 1 0 A) [0.6 0.4] B) C) 0.8 0.2 0 1 0.35 0.65

D) [0.66 0.34] 22) D) [0.636 0.364]

2 3 23) Find the limiting matrix P corresponding to the transition matrix P = 1 4 Round to the nearest thousandths. A) 1 0 0 1 C) 0.429 0.571 0.429 0.571 24)

1 3 3 . 4

23)

B) [0.429 0.571] D) 0.528 0.472 0.354 0.646

A B C D A 0 0 1 0 Identify the absorbing state(s) in the transition matrix P = B 0 1 0 0 C 0 1 0 0 D 0 0 0 1 A) A and D B) B and D C) C and D

24)

D) A and B

25) The probability that an assembly line operation works correctly depends on whether it worked correctly the last time it was used. There is a 0.91 chance that the line will work correctly if it worked correctly the time before and a 0.68 chance that it will work correctly if it did not work correctly the time before. After setting up a transition matrix with this information, find the long-run probability that the line will work correctly. A) [0.117 0.883] B) [0.883 0.117] C) [0.802 0.198] D) [0.883 0.883]

25)

26) Given the transition matrix: A B C D A 0.4 0.3 0.2 0.1 1 0 0 P= B 0 C 0 0 1 0 D 0.2 0.5 0.1 0.2

26)

P4 =

A B C D

A B C D 0.0396 0.5942 0.3518 0.0144 0 1 0 0 0 0 1 0 0.0288 0.756 0.2044 0.0108

Find the probability of going from state D to state A in four trials. A) 0.0288 B) 0.3518 C) 0

D) 0.0396

27) Given the transition matrix: A B C D A 0.4 0.3 0.2 0.1 1 0 0 P= B 0 C 0 0 1 0 D 0.2 0.5 0.1 0.2

27)

Find P 4. A) A B C D A 0.0396 0.5942 0.3518 0.0144 0 1 0 0 P= B C 0 0 1 0 D 0.0288 0.756 0.2044 0.0108 B) A B C D A 0.4 0.3 0.2 0.1 1 0 0 P= B 0 C 0 0 1 0 D 0.2 0.5 0.1 0.2 C) A B C D A 0.18 0.47 0.29 0.06 0 1 0 0 P= B C 0 0 1 0 D 0.12 0.66 0.16 0.06 D) A B C D A 0.084 0.554 0.332 0.03 0 1 0 0 P= B C 0 0 1 0 D 0.06 0.726 0.19 0.024 28) Find all absorbing states for the transition matrix, and indicate whether or not the matrix is that of an absorbing Markov chain. 1 2 3 1 0.9 0 0.1 2 0 1 0 3 0.6 0 0.4 A) State 2 is absorbing; matrix is not an absorbing Markov chain. B) State 3 is absorbing; matrix is not an absorbing Markov chain. C) State 3 is absorbing; matrix is an absorbing Markov chain. D) State 2 is absorbing; matrix is an absorbing Markov chain.

28)

29) Laurinburg is experiencing a population movement out of the city to the suburbs. Currently 85% of the total population live in the city with the remaining 15% living in the suburbs. It has been shown that each year 7% of the city residents move to the suburbs, while only 1% of the suburb population move back to the city. Assuming population remains constant for both, what percent of the total will remain in the city after 5 years. Express your answer rounded to hundredths of a percent. A) 60.28% B) 64.44% C) 39.72% D) 35.56% 0.80 0.10 0.10 30) Find the stationary matrix for the transition matrix P = 0.15 0.80 0.05 0.20 0.70 0.10 Round the numbers in your answer to the nearest hundredth. A) [1 0 0] B) [0.44 0.48 0.08] C) [0.33 0.40 0.27] D) [0.33 0.67 0]

29)

30)

31) Find a standard form for the absorbing Markov chain with the transition matrix A B C D A 1/4 1/4 1/4 1/4 B 0 1 0 0 C 1/2 0 1/2 0 D 0 0 0 1 A) A B C D B) B D A C B 1 0 0 0 B 1 0 0 0 D 0 1 0 0 D 0 1 0 0 A 1/4 1/4 1/4 1/4 A 1/4 1/4 1/4 1/4 C 0 0 1/2 1/2 C 0 0 1/2 1/2 C) D B C A D) A B C D B 1 0 0 0 B 1 0 0 0 D 0 1 0 0 D 0 1 0 0 A 1/4 1/4 1/4 1/4 A 1/4 1/4 1/4 1/4 C 0 0 1/2 1/2 C 0 0 1/2 1/2

31)

32) Find the fundamental matrix F for the absorbing Markov chain with the given matrix. Express your answer in fraction form. 1 0 0 0 1 0 3 4 1 7 7 7

32)

7 A) F = 4

7 B) F = 3

C) F = [3]

7 D) F = 6

Solve the problem. 33) Weather is classified as sunny or cloudy in a certain place. What are the long-term predictions for sunny and cloudy days? Round numbers to the nearest thousandths.

33)

Sunny Cloudy Sunny Cloudy

0.9

0.1

0.6

0.4

A) 0.855 0.145 C) 0.900 0.100

B) 0.143 0.857 D) 0.857 0.143

Provide an appropriate response. 34) The transition matrix for a Markov process is: State A B A 0.1 0.9 = P State B 0.3 0.7

34)

Find P 2. A) 0.244 0.756 0.252 0.748 C) 0.28 0.72 0.24 0.76

B) 0.1 0.9 0.3 0.7 D) 0.72 0.28 0.36 0.64

Construct the transition matrix that represents the data. 35) If it snows today, there is a 70 percent chance of snow tomorrow; however if it does not snow today, there is a 40 percent chance that it will not snow tomorrow. A) B) C) D) 0.7 0.3 0.6 0.4 0.7 0.3 0.4 0.6 0.6 0.4 0.7 0.3 0.4 0.6 0.7 0.3 Provide an appropriate response. 36) The transition matrix for a Markov process is: State A B 0.7 = P State A 0.3 B 0.9 0.1

35)

36)

Find the second state matrix if the initial state is S 0 = 0.3 0.7 . A) [0.3 0.7] B) [0.6192 0.3808] C) [ 0.468 0.538] D) [0.72 0.28] 37)

A B C A 0.2 0.3 0.5 Using a graphing utility to compute powers of P = B 0.1 0.8 0.1 , find the smallest n such C 0.4 0.3 0.3 that the corresponding entries in P n and P n+1 are, when rounded to 3 decimal places, equal. A) 11

B) 6

C) 2 9

D) 3

37)

38) Fayetteville is experiencing a population movement out of the city to the suburbs. Currently 85% of the total population live in the city with the remaining 15% living in the suburbs. It has been shown that each year 7% of the city residents move to the suburbs, while only 1% of the suburb population move back to the city. Assuming population remains constant for both, what percent of the total will remain in the suburbs after 5 years. Express your answer rounded to hundredths of a percent. A) 35.56% B) 64.44 C) 60.28% D) 39.72%

38)

39) Decide whether or not the transition matrix is regular. Answer Yes or No. 0.61 0 0.39 0.47 0.53 0 0 0.24 0.76 A) Yes B) No

39)

Construct the transition matrix that represents the data. 40) 10 percent of the people in one generation who have a certain physical characteristic will pass that characteristic on to the next generation. 40 percent of the people in one generation who do not have this characteristic will pass it on to the next generation. A) B) C) D) 0.6 0.4 0.1 0.9 0.9 0.1 0.4 0.6 0.9 0.1 0.4 0.6 0.6 0.4 0.1 0.9 Provide an appropriate response. 41) A red urn contains 4 red marbles, 2 blue marbles, and 4 green marbles. A blue urn contains 2 red marbles, 2 blue marbles, and 1 green marble. A green urn contains 3 green marbles. A marble is selected from an urn, the color is noted, and the marble is returned to the urn from which it was drawn. The next marble is drawn from the urn whose color is the same as the marble just drawn. Thus, this is a Markov process with three states: draw from the red urn, draw from the blue urn, or draw from the green urn.

40)

41)

Find the limiting matrix P, if it exists, and describe the long-run behavior of this process. A) B) G B R G B R G 1 0 0 G 1 0 0 P = B 0 0.1 0.4 P = B 0.2 0.4 0.4 R 1 0 0 R 1 0 0 C) D) G BR G B R G 1 0 0 G 1 0 0 P= B 1 0 0 P = B 0.2 0.4 0.4 R 1 0 0 R 0.4 0.2 0.4 42) Find the stationary matrix for the transition matrix P = 0.1 0.9 . 0.6 0.4 A) [0.6 0.4] B) [0.4 0.6] C) 0.1 0.9 0.6 0.4

42) D) [0.16 0.36]

Solve the problem. 43) The probability that an assembly line works correctly depends on whether the line worked correctly the last time. Find the probability that the line will work in the long run. Round your answer as appropriate.

43)

Works Does not works

0.9

0.1

doesn't

0.6

0.4

A) 0.900

B) 0.857

C) 0.855

D) 0.143

Provide an appropriate response. 44) Among patients who were improving on a certain day (in the critical care unit of a certain hospital), the following was determined to be true on the next day: 58% were still improving; 10% were stable; 7% were deteriorating; and 25% had been discharged. Among the patients who were stable on a certain day, the following was determined to be true on the next day: 34% were improving; 44% were still stable; 18% were deteriorating; 3% had been discharged; and 1% had died. Among the patients who were deteriorating on a certain day, the following was determined to be true on the next day: 13% were improving; 42% were stable; 41% were still deteriorating; none had been discharged; and 4% had died. What is the expected number of additional days that a patient, who is improving on that certain day, will spend in the critical care unit? Round your answer to the nearest hundredth. A) 3.22 days B) 4.65 days C) 7.53 days D) 5.76 days

44)

45) Suppose that for a certain absorbing Markov chain the fundamental matrix is found to be $1 $2 $3 $1 1.5 1.0 0.5 $2 1.0 2.0 1.0 $3 0.5 1.0 1.5 What is the expected number of times a person will have $3, given that he started with $2? A) 0.5 B) 1.0 C) 1.5 D) 2.0

45)

46) Find a standard form for the absorbing Markov chain with the following transition matrix : A B C D A 0.4 0.2 0.1 0.3 B 0 1 0 0 C 0 0.2 0.1 0.7 D 0 0 0 1 A) B) B D C A B D C A B 1 0 0 0 A 1 0 0 0 D 0 1 0 0 B 0 1 0 0 C 0.2 0.7 0.1 0 C 0.2 0.7 0.1 0 A 0.2 0.3 0.1 0.4 D 0.2 0.3 0.1 0.4 C) D) A B C D B D C A B 1 0 0 0 B 1 0 0 0 D 0 1 0 0 D 0 1 0 0 C 0.2 0.7 0.1 0 C 0.2 0.7 0.1 0 A 0.2 0.3 0.1 0.4 A 0.3 0.2 0.4 0.1

46)

47) Dublin is experiencing a population movement out of the city to the suburbs. Currently 85% of the total population live in the city with the remaining 15% living in the suburbs. It has been shown that each year 7% of the city residents move to the suburbs, while only 1% of the suburb population move back to the city. Assuming population remains constant for both, what percent of the total will remain in the city after 2 years. Express your answer rounded to hundredths of a percent. A) 31.05% B) 26.14% C) 73.86% D) 79.2% 1 2 48) Find the limiting matrix P corresponding to the transition matrix P = 3 7

47)

1 2 4 . 7

48)

Round to the nearest thousandths. 3 7

4 7

A) [0.462 0.538]

C) [0.5 0.5]

D) 0.462 0.538 0.462 0.538

49)

For the transition matrix

, find the probability that if one starts in state B, one

will end up in state A over the long run. 4 3 A) B) 5 5

9 5

50) Decide whether or not the transition matrix is regular. Answer Yes or No. 1 0 0 0 0 1 0 1 0 A) Yes B) No

D) 1

50)

51) Find a standard form for the absorbing Markov chain with the transition matrix A B C A 0 0 1 B 0 1 0 C 0.2 0.6 0.2 A) B) B A C B A C B 1 0 0 A 1 0 0 A 0 0 1 B 0 0 1 C 0.6 0.2 0.2 C 0.6 0.2 0.2 C) A B C D) A B C B 1 0 0 A 0 0 1 A 0 0 1 B 0 1 0 C 0.6 0.2 0.2 C 0.2 0.6 0.2 Solve the problem. 52) Rats are kept in a cage with two compartments (A and B). Rats in A move to B with probability 0.7. Rats in B move to A with probability 0.2. Find the long-term trend for rats in each compartment. Round numbers to the nearest thousandth. A) 0.778 0.222 B) 0.300 0.700 C) 0.285 0.715 D) 0.222 0.778

51)

52)

Provide an appropriate response. 53) For the transition matrix P = 0.36 0.64 find P exactly by converting P 16 to fraction form. 0.20 0.80 5 16 21 21 A) 0.213 0.698 B) 5 16 0.218 0.715 21 21

5 21

16 21

54) Given the transition matrix: A B C D A 0.4 0.3 0.2 0.1 1 0 0 P= B 0 C 0 0 1 0 D 0.2 0.5 0.1 0.2

53)

D) [0.213 0.698]

54)

P4 =

A B C D

A B C D 0.0396 0.5942 0.3518 0.0144 0 1 0 0 0 0 1 0 0.0288 0.756 0.2044 0.0108

Find the probability of going from state A to state D in four trials. A) 0.0144 B) 0 C) 0.0396

D) 0.3518

55) Find the fundamental matrix F for the absorbing Markov chain with the given matrix. Express your answer in fraction form. 1 0 0 0 1 0 0.12 0.72 0.16 21 11 4 25 A) F = 25 B) F = 9 C) F = 25 D) F = 21

55)

56) According to data collected during one year in a large metropolitan community, 30% of commuters used public transportation to get to work, and this rose by 4% the following year. This is modeled by the transition matrix

56)

′

P P P 0.9 0.1 M= ′ P 0.1 0.9

S 0 = [0.3 0.7] ′

where P represents the percentage of people that use public transportation and P the percentage of people that do not. What percentage of commuters in this community will use public transportation in the long run? A) 34% B) 30% C) 37.2% D) 50% 57) Find all absorbing states for the transition matrix, and indicate whether or not the matrix is that of an absorbing Markov chain. 1 2 3 4 1 0.1 0 0.1 0.8 2 0 0.1 0 0 3 0.1 0.2 0.3 0.4 4 0 0 0 1 A) State 2 is an absorbing; the matrix is not an absorbing Markov chain. B) State 2 and 3 are absorbing; the matrix is an absorbing Markov chain. C) State 2 and 3 are absorbing; the matrix is not an absorbing Markov chain. D) State 2 is an absorbing; the matrix is an absorbing Markov chain.

57)

A small town has only two dry cleaners, Fast and Speedy. Fast hopes to increase its market share by conducting an extensive advertising campaign. The initial market share for Fast was 40% and 60% for Speedy. Solve the problem. 58) Find the probability that a customer using Fast initially will use Fast for his third batch of 58) clothes. Use the following transition matrix. Round your answer to the nearest hundredth. Fast Speedy Fast

0.62

0.38

Speedy 0.70

0.30

A) 0.35

B) 0.65

C) 0.64

D) 0.36

Provide an appropriate response. 59)

A B C D A 0.4 0.1 0.4 0.1 1 0 0 , find the limiting matrix. Use fractional entries. For the transition matrix B 0 C 0.3 0.1 0.2 0.4 D 0 0 0 1 A) B) A B C D B A C D 2 1 2 A 1 0 0 B 0 0 3 3 3 B C

D C)

1 1 4

B C D 1 2 0 3 3

0 A 0 B C

3 4

0 3 4

A 0 C 0

1 1 4

D 0

B 1 3

D 2 3

B 0 C 0

1 1 4

D 0

1 1 4

A 0

60) The transition matrix for a Markov process is: State A B A 0.3 0.7 = P State B 0.9 0.1 Find P 2. A) 0.72 0.36 C) 0.72 0.28

59)

0 3 4

0 3 4 1

60)

0.28 0.64 0.36 0.64

B) 0.64 0.28 0.36 0.72 D) 0.468 0.532 0.684 0.316

Answer Key Testname: CHAP 09_14E

1) A 2) B 3) B 4) C 5) C 6) C 7) D 8) C 9) A 10) C 11) A 12) B 13) C 14) C 15) A 16) D 17) D 18) A 19) A 20) C 21) A 22) D 23) C 24) B 25) B 26) A 27) A 28) A 29) A 30) B 31) B 32) D 33) D 34) C 35) A 36) C 37) A 38) D 39) A 40) B 41) C 42) B 43) B 44) D 45) B 46) A 47) C 48) D 49) B 16

Answer Key Testname: CHAP 09_14E

50) B 51) A 52) D 53) B 54) A 55) D 56) D 57) B 58) B 59) D 60) A

CHAPTER 10

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct a broken-line graph of the data in the table. 1) The following table gives the total amount of precipitation during the given months. Use time on the horizontal scale for your line graph. Total Precipitation, Month in Inches Nov. 1.52 Dec. 2.72 Jan. 3.76 Feb. 6.04 Mar. 5.56 April 7.44 May 8.28 Construct the specified histogram. 2) Construct a histogram for the following frequency table. Class Interval 0.5-1.5 1.5-2.5 2.5-3.5 3.5-4.5 4.5-5.5

Frequency 6 21 16 4 3

Construct a histogram for the binomial distribution P(x) = Cn, xpxqn - x , and compute the mean and standard deviation. 3) n = 6, p = 3 10

Construct the specified histogram. 4) Eighty U. S households were surveyed about MP3 players. The table gives the frequency distribution for the data.

# of MP3's Frequency 1 5 2 10 3 30 4 25 5 10 50 40 30 20 10

Construct a histogram. A fair coin is tossed fourteen times. What is the probability of obtaining the following? Express the answer both in terms of Cn, k and as a four-place decimal. 5) Exactly 12 heads?

Construct a frequency table. 6) The following are the heights, in inches, of ten middle school basketball players. Determine the data range for the following data set. Also, construct a frequency table for the data set, use a class interval width of 2 and start with 62.5 inches.

72 70 68 67 66 67 63 71 65 74 Construct a broken-line graph of the data in the table. 7) The following table shows the number of computer sales made at Computer Buy over five months. Use time on the horizontal scale for your line graph. Number of Month Computers Sold 1 213 2 256 3 311 4 527 5 489

Construct the specified histogram. 8) Construct a histogram for the following set of data on oil rig utilization by type.

Oil Rig Type Jackup Semisub Drill Ship Submersible

Percentage of Rigs Currently Utilized (Feb. 2004) 86% 74% 82% 86%

Provide an appropriate response. 9) A botanist wants to grow a rare plant in his greenhouse. The probability that a given bulb will mature is 0.42. Suppose 6 bulbs are planted. (A) Write the probability function defining this distribution. (B) What is the probability that 3 or more bulbs will mature? (Round your answer to three decimal places.)

Construct a histogram for the binomial distribution P(x) = Cn, xpxqn - x , and compute the mean and standard deviation. 10) n = 7 and p = 1 2

10)

Construct the specified histogram. 11) Twenty voters were asked their age. The results are summarized in the frequency table below.

11)

Age of Number of voters voters 20- 30 5 30- 40 5 40- 50 6 50- 60 0 60- 70 4

Construct a histogram. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 12) According to a college survey, 22% of all students do not work. Find the standard deviation for the random variable X, the number of students who do not work in samples of size 16. A) 3.52 B) 2.75 C) 1.66 D) 1.88 3

12)

13) What proportion of the following sample of ten measurements lies within 1 standard deviation of the mean? 5 6 5 4 6 6 7 8 7 6 A) 60% B) 40% C) 80% D) 100%

13)

14) Following is a sample of the percent increases in the price of a house from 2002 to 2007 in 8 regions of the U. S. 75 130 145 150 150 225 225 300 Find the mean. A) 175 B) 225 C) 300 D) 150

14)

15) The test scores of 40 driver license applicants are summarized in the frequency table below. Find the standard deviation.

15)

Score Students 50 - 59 8 60 - 69 7 70 - 79 12 80 - 89 7 90 - 99 6 Round your answer to one decimal place. A) 12.1 B) 13.4 C) 14.1

D) 12.7

E) 12.0

16) A normal random variable X has mean 40 and standard deviation 16. Find the area under the normal curve above the interval 16-60. A) 0.8089 B) 0.8276 C) 0.7324 D) 0.8962

16)

17) Find the median for the following grouped data.

17)

Interval 2.5-4.5 4.5-6.5 6.5-8.5 8.5-10.5 10.5-12.5 A) 9.5

Frequency 1 8 4 5 3 B) 11.5

C) 7.25

D) 7.60

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. 18) A certain question on a drivers test is answered correctly by 22% of the respondents. Estimate 18) the probability that among the next 150 responses there will be at most 40 correct answers. A) 0.8997 B) 0.9306 C) 0.1003 D) 0.0694

Construct a frequency polygon. 19) Weight of bag Number of of chips (in ounces) bags 15.6 -15.7 6 15.8 -15.9 22 16.0 -16.1 35 16.2-16.3 27 16.4-16.5 10

19)

A) Frequency

Weight in ounces B) Frequency

Weight in ounces Provide an appropriate response. 20) According to a college survey, 22% of all students work full time. Find the mean for the random variable X, the number of students who work full time in samples of size 16. Find the mean of the binomial distribution. A) 4.00 B) 0.22 C) 3.52 D) 2.75

20)

Given a normal distribution with mean 120 and standard deviation 5, find the number of standard deviations the measurement is from the mean. Express the answer as a positive number. 21) 134.9 21) A) 2.18 B) 3.02 C) 3.25 D) 2.98 Provide an appropriate response. 22) Here are the commutes (in miles) for a group of six students. Find the standard deviation. 14.7 16.3 34.0 33.7 22.6 16.0 Round to two decimal places. A) 3540.25 B) 8.93 C) 3141.91 D) 33.93 5

22)

23) If a baseball player has a batting average of 0.420, what is the probability that the player will get at least 2 hits in the next four times at bat? A) 0.042 B) 0.333 C) 0.50 D) 0.559 Construct a pie graph, with sectors given in percent, to represent the data in the given table. 24) Age Number of people 25-35 294 35-45 396 45-55 348 55-65 186 65-75 90

24)

B) 90%

186%

294%

14%

22%

348%

396%

27%

30%

D) 7%

10%

19%

22%

14%

24%

22%

30%

27%

26%

23)

Which single measure of central tendency - mean, median, or mode - would you say best describes the given set of measurements? 25) 2.02 1.93 2.24 3.56 1.98 25) 2.05 1.97 2.20 3.56 12.5 A) The Mean B) The Mode C) The Median Provide an appropriate response. 26) What proportion of the following sample of ten measurements lies within 2 standard deviations of the mean? 1 5 9 2 6 3 3 4 5 2 A) 90% B) 70% C) 80% D) 100%

26)

27) The life expectancy (in hours) of a fluorescent tube is normally distributed with mean 7,000 and standard deviation 1,000. Find the probability that a tube lasts for more than 8,900 hours. A) 0.0287 B) 0.9719 C) 0.9713 D) 0.0281

27)

28) Find the mean for the data set: 2, 11, 35, 2, 9, 35, 11, 9, 7, 2, 2, 2, 2, 9, 2 A) 11 B) 2

28) C) 9.33

D) 7

Assume the distribution is normal. Use the area of the normal curve to answer the question. Round to the nearest whole percent. 29) A machine produces screws with an average diameter of 0.30 inches and a standard deviation of 29) 0.01 inches. What is the probability that a screw will have a diameter greater than 0.32 inches? A) 3% B) 1% C) 2% D) 97% Evaluate Cn,x pxqn-x for the given values of n, x, and p. 30) n = 6, x = 3, p = 1 6 A) 0.0286

30) B) 0.0322

C) 0.0536

D) 0.0154

Construct a pie graph, with sectors given in percent, to represent the data in the given table. 31) Favorite Beverage Number of responses Cola 340 Juice 210 Milk 230 Tea 310 Water 150

31)

150%

5% 340%

32%

310%

28% 210%

16%

230%

19%

12% 24%

27%

28%

25% 21%

17%

19%

Given a normal distribution with mean 120 and standard deviation 5, find the number of standard deviations the measurement is from the mean. Express the answer as a positive number. 32) 114.2 32) A) 2.4 B) 1.2 C) 2.12 D) 1.16 Construct a frequency table. 33) The following is the number of hours students studied per week on average. Use five intervals, starting with 0 - 4. 2 5

9 13 A)

14 19 17 24

20 21 15 13

18 10 5 4 9 14

2 19 B)

Interval Frequency 0-4 3 5-9 4 10-14 4 15-19 6 20-24 3

Interval Frequency 0-4 3 5-9 4 10-14 5 15-19 5 20-24 3

D) Interval Frequency 0-4 3 5-9 4 10-14 5 15-19 4 20-24 4

Interval Frequency 0-4 3 5-9 3 10-14 6 15-19 5 20-24 3

33)

Provide an appropriate response. 34) Here are the commutes (in miles) for a group of six students. 14.7 16.3

34.0

33.7

22.6

34)

16.0

Find the range rounded to one decimal place. A) 16.0 B) 19.5

C) 34.0

D) 19.3

Given a normal distribution with mean 120 and standard deviation 5, find the number of standard deviations the measurement is from the mean. Express the answer as a positive number. 35) 107 35) A) 3.2 B) 2.6 C) 2 D) 1.4 Provide an appropriate response. 36) Following is a sample of the percent increases in the price of a house from 2000 to 2005 in 8 regions of the U. S. 75 130 145 150 150 225 225 300 Find the median. A) 150 B) 300 C) 137.5 D) 225

36)

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. 37) Two percent of flat irons produced at a certain plant are defective. Estimate the probability that 37) of 10,000 randomly selected flat irons, the number of defectives is between 195 and 210 inclusive. A) 0.4034 B) 0.3989 C) 0.4251 D) 0.4017 Provide an appropriate response. 38) The register of a college recorded the amount of time each student spent waiting in line during peak registration hours one Monday. The frequency table below summarizes the results. Find the standard deviation. Round your answer to one decimal place. Waiting time Number of (minutes) students 0-3 15 4-7 15 8 - 11 8 12 - 15 9 16 - 19 0 20 - 23 3 A) 5.4 B) 5.8

C) 5.9

D) 5.3

38)

E) 5.6

39) A test consists of 10 true/false questions. To pass the test a student must answer at least 7 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test? A) 0.172 B) 0.117 C) 0.945 D) 0.055

39)

Evaluate Cn,x pxqn-x for the given values of n, x, and p. 40) n = 30, x = 12, p = 0.20 A) 0.1082

40) B) 0.0028

C) 0.0064

D) 0.0139

Provide an appropriate response. 41) Find the mode for the data set: 2, 11, 35, 2, 9, 35, 11, 9, 7, 2, 2, 2, 2, 9, 2 A) 9.3 B) 7

41) C) 11

D) 2

42) Find the mean for the following grouped data. Interval 5.5 -9.5 9.5-13.5 13.5-17.5 17.5-21.5 21.5-25.5 A) 15.5

42)

Frequency 3 1 5 7 4 B) 7.5

C) 19.5

D) 17.1

43) The frequency distribution below gives the weight in grams of 100 laboratory mice. What is the probability that the weight of a healthy mouse selected randomly from the sample will be more than 52 grams?

43)

Weight in grams of Laboratory Mice Class Interval Frequency 43.5-46.5 16 46.5-49.5 24 49.5-52.5 27 52.5-55.5 24 55.5-58.5 9 A) 0.23

B) 0.50

C) 0.33

D) 0.67

44) In the English department of a midwestern university, the annual salaries of five faculty members are $34,000, $35,000, $36,000, $36,500 and $65,000. Compute the median. A) $41,300 B) $50,750 C) $34,000 D) $36,000 Construct a pie graph, with sectors given in percent, to represent the data in the given table. 45) Favorite Restaurant Style Number of Responses Chinese 207 Indian 135 Mexican 216 Thai 162

44)

45)

23%

30%

29%

19%

22.5%

28.8%

30%

18.7%

162%

216%

207%

135%

21%

32%

31%

20%

Provide an appropriate response. 46) Here are the prices for 8 different MP3 players. Find the standard deviation. $195 $358 $201 $276 $161 Round to one decimal place. A) 329.5 B) 238.5

$301

$387

$128 C) 144.5

47) Find the standard deviation for the following data set: 2, 2, 2, 5, 5, 6, 6, 8, 8, 9 A) 2.49 B) 2 C) 5 48) Find the median for the data set: 2, 14, 35, 2, 8, 35, 14, 8, 6, 2, 2, 2, 2, 8, 2 A) 6 B) 8

46)

D) 94.1 47) D) 6.21 48)

C) 9.47

D) 14.25

49) Given the frequency distribution below, what is the probability of the hourly wage of a person chosen at random from the sample being less than $4.495?

49)

Hourly Wage of Part-time Fast Food Service Workers

A) 0.08

B) 0.75

C) 0.20

D) 0.80

Construct a bar graph of the given frequency distribution. 50) The frequency distribution indicates the height in feet of persons in a group of 216 people. Height (feet) Number of persons 4-5 24 5-6 136 6-7 48 7-8 8

50)

A) 320

160

B) 160

C) 320

160

D) 160

51) The frequency distribution indicates the number of fish caught by each fisherman in a group of 250 fishermen. Number of Number of fish caught persons 1 80 2 60 3 50 4 10 5 30 6 20

51)

A) 80

0 B) 80

0 C) 40

D) 80

Provide an appropriate response. 52) A small company employs a supervisor at $1200 a week, an inventory manager at $800 a week, 5 stock boys at $400 a week each, and 3 drivers at $700 a week each. A) $1260 B) $550 C) $1017 D) $610 53) Find the mean for the following grouped data. Interval 9.5 -12.5 12.5-15.5 15.5-18.5 18.5-21.5 A) 14

52)

53)

Frequency 2 3 8 4 B) 11

C) 16.47

D) 17

54) The probability that a daisy seed will germinate is 0.7. A gardener plants seeds in batches of 12. Find the mean for the binomial random variable X, the number of seeds germinating in each batch. A) 10.8 B) 8.5 C) 3.6 D) 8.4

54)

55) Find the mode for the data set: 3, 13, 30, 3, 8, 30, 13, 8, 6, 3, 3, 3, 3, 8, 3 A) 7 B) 13

55) C) 3

D) 9.6

56) Find the median for the following grouped data. Interval 7.5-8.5 8.5-9.5 9.5-10.5 10.5-11.5 11.5 -12.5 A) 9.875

56)

Frequency 2 7 4 5 3 B) 8.35

C) 10

D) 11.35

Use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with given the properties. 57) n = 20, p = .8 57) A) Yes B) No Provide an appropriate response. 58) Find the standard deviation for the following grouped data: Interval 9.5-12.5 12.5-15.5 15.5-18.5 18.5-21.5

58)

Frequency 3 2 7 4

A) 7.56

B) 3.09

C) 3.256

D) 9.5625

Assume the distribution is normal. Use the area of the normal curve to answer the question. Round to the nearest whole percent. 59) The average size of the bass in a lake is 11.4 inches, with a standard deviation of 3.2 inches. Find 59) the probability of catching a bass longer than 17 inches. A) 8% B) 96% C) 4% D) 5% Given a normal distribution with mean 120 and standard deviation 5, find the number of standard deviations the measurement is from the mean. Express the answer as a positive number. 60) 125 60) A) 3 B) 0 C) 2 D) 1 Provide an appropriate response. 61) Here are the prices for 8 different MP3 players. Find the range. $195

$358

A) 230

$201

$276

$161

$301

$387

B) 195

61)

$128 C) 200

D) 259

62) A normal distribution has mean 200 and standard deviation 50. Find the area under the normal curve from the mean to 224. A) 0.6808 B) 0.6844 C) 0.5000 D) 0.1844

62)

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. 63) In one county, the conviction rate for DUI is 85%. Estimate the probability that of the next 100 63) DUI summonses issued, there will be at least 90 convictions. A) 0.1038 B) 0.0420 C) 0.8962 D) 0.3962 Use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with given the properties. 64) n = 400, p = 0.06 64) A) No B) Yes

Construct a frequency polygon. 65) Life of bulb Number (in hours) of bulbs 400-499 45 500 -599 80 600 -699 120 700 -799 70 800 -899 35

65)

A) Frequency

Hours of bulb life B) Frequency

Hours of bulb life Provide an appropriate response. 66) If an investor purchased 50 shares of Pearson Education stock at $85 per share, 90 shares at $105 per share, 120 shares at $110 per share and another 75 shares at $130 per share. What is the mean cost per share? A) $130.50 B) $105.50 C) $109.40 D) $110.25 67) In the English department of a midwestern university, the annual salaries of five faculty members are $34,000, $35,000, $36,000, $36,500 and $65,000. Compute the mean. A) $41,300 B) $36000 C) $50,750 D) $34,000

66)

67)

Which single measure of central tendency - mean, median, or mode - would you say best describes the given set of measurements? 68) 36 41 72 81 64 68) 52 83 78 71 40 A) The Median B) The Mode C) The Mean 18

Assume the distribution is normal. Use the area of the normal curve to answer the question. Round to the nearest whole percent. 69) The mean clotting time of blood is 7.35 seconds, with a standard deviation of 0.35 seconds. What 69) is the probability that blood clotting time will be less than 7 seconds? A) 16% B) 15% C) 14% D) 84% Provide an appropriate response. 70) In a certain college, 33% of the math majors belong to foreign student. If 10 students are selected at random from the math majors, that is the probability that no more than 6 are foreign? A) 0.9846 B) 0.913 C) 0.9815 D) 0.0547

70)

Answer Key Testname: CHAP 10_14E

1) Answers may vary. A possible answer follows.

9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0

Answer Key Testname: CHAP 10_14E

3) ! = 1.8; σ = 1.12

4) 50 40 30 20 10

5) C14, 12(0.5) 14 ≈ 0.0056 6) Data range: 11 Class Interval Tally 62.5 -64.5 ∣ 64.5 -66.5 ∣∣ 66.5 -68.5 ∣∣∣ 68.5 -70.5 ∣ 70.5 -72.5 ∣∣ 72.5 -74.5 ∣

Frequency 1 2 3 1 2 1

Relative Frequency 0.1 0.2 0.3 0.1 0.2 0.1

Answer Key Testname: CHAP 10_14E

7) Answers may vary. A possible answer follows.

600 500 400 300 200 100 0

9) (A) P(x) = C6, x (0.42) x(0.58) 6-x (B) P(3 or more successes) = 0.497

Answer Key Testname: CHAP 10_14E

10) ! = 3.5; σ = 1.32

11)

12) C 13) C 14) A 15) B 16) B 17) C 18) B 19) B 20) C 21) D 22) B 23) D 24) B 25) C 26) A 27) A 28) C 29) C 23

Answer Key Testname: CHAP 10_14E

30) C 31) D 32) D 33) B 34) D 35) B 36) A 37) C 38) E 39) A 40) C 41) D 42) D 43) C 44) D 45) A 46) D 47) A 48) A 49) D 50) B 51) A 52) D 53) C 54) D 55) C 56) A 57) B 58) B 59) C 60) D 61) D 62) D 63) A 64) B 65) B 66) C 67) A 68) C 69) A 70) C

CHAPTER 11

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Daisy and Gus write down one of the numbers 1, 4 , or 7. If the sum of the numbers is even Daisy pays Gus that number of dimes. If the sum of the numbers is odd, Gus pays Daisy that number of dimes. Write Daisy's game matrix that corresponds to this situation. A) 1 4 7 B) 1 4 7 1 2 5 8 1 -2 5 -8 4 5 8 11 4 5 -8 11 7 8 11 14 7 -8 11 -14 C) 1 4 7 D) 1 4 7 1 2 5 8 1 -2 -5 -8 4 5 8 11 4 -5 -8 11 7 8 11 -14 7 -8 11 -14 1 2) Find the expected value of the game matrix A = 0 0 if player 1 and player 2 decide on strategies 1 3 1 1 1 P= 3 3 3

1 and Q = 3

0 1 0

0 0 1

2 3

1 3 A)

1 3

8 3

D) 1

3) Suppose a matrix game has the following nonstrictly determined matrix: M= 4 1 3 5 Set up (but do not solve), the two corresponding linear programming problem used to solve this matrix game. A) Minimize y = x1+ x2 B) Maximize y = x1+ x2 subject to

Maximize

subject to 4x1+ 3x2 ≥ 1 x1+ 5x2 ≥ 1

4x1+ 3x2 ≥ 1 x1+ 5x2 ≥ 1

x1, x2 ≥ 0 y = z1+ z2

Minimize

subject to

subject to 4z1+ z2 ≤ 1

4z1+ z2 ≤ 1

3z1+ 5z2 ≤ 1

z1, z2 ≥ 0

Determine if the statement is true or false. 4) If a matrix game is fair, then both players have optimal strategies that are pure. A) False B) True

Provide an appropriate response. 5) Find the expected value of the game matrix A = 4 0 2 3 1 2 1 1 strategies P = 2 2 and Q = 1 .

if player 1 and player 2 decide on

2 A)

9 4

1 2

4 9

4 5

6) Is the following matrix game strictly determined? Answer Yes or No. -2 4 -3 6 A) Yes B) No

7) Determine which row(s) and column(s) of the game matrix are recessive.

4 -3 1 -2 4 -4 0 -2 -1 2 3 2 0 0 5 2 A) Recessive row: R 2; recessive column: C3 , C4 B) Recessive row: R 2; recessive column: C4 C) Recessive row: R 2; recessive column: C3 D) Recessive row: R 1; recessive column: C3 , C4 Find the smallest integer k ≥ 0 such that adding k to each entry of the given matrix produces a matrix with all positive payoffs. 8) 0 1 8) 2 4 A) 0 B) 1 C) 2 D) 3

Solve the problem. 9) A person is considering three different stocks, and each is sensitive to a certain economic indicator. The indicator will be positive, neutral, or negative, and fluctuate randomly. The payoffs are given in the table below, in thousands of dollars. What investment strategy should the person make to obtain the best expected value of profit?

Payoff (in thousands of dollars) Positive Neutral Negative Stock A 6 14 13 Stock B 6 7 8 Stock C 15 10 12 A) Invest in Stock A with probability 0, invest in Stock B with probability 1, and invest in Stock C with probability 0. B) Invest in Stock A with probability 5/13, invest in Stock B with probability 0, and invest in Stock C with probability 8/13. C) Invest in Stock A with probability 1/3, invest in Stock B with probability 1/3, and invest in Stock C with probability 1/3. D) Invest in Stock A with probability 1, invest in Stock B with probability 0, and invest in Stock C with probability 0. Provide an appropriate response. 10) Find the saddle values, if it exists, for the matrix game (include row and column location). 3 0 -2 -1 2 -3 0 -1 4 2 1 0 A) 0; row 3 column 4 B) 3; row 1 column 1 C) 0; row 1 column 2 D) 4; row 3 column 1 Solve the problem. 11) A person has hired an investment broker to buy stock. The broker has three different stock funds that are of interest, but each is sensitive to a certain economic indicator that is impossible to predict. The indicator will be positive, neutral, or negative. The table below shows the payoffs in thousands of dollars. Find the strategy that the broker should recommend to maximize the expected value of the investment. Payoff (in thousands of dollars) Positive Neutral Negative Stock A 9 17 16 Stock B 18 13 15 Stock C 9 10 11 A) Invest in Stock A with probability 4/13, invest in Stock B with probability 9/13, and invest in Stock C with probability 0. B) Invest in Stock A with probability 5/13, invest in Stock B with probability 8/13, and invest in Stock C with probability 0. C) Invest in Stock A with probability 0, invest in Stock B with probability 1, and invest in Stock C with probability 0. D) Invest in Stock A with probability 1, invest in Stock B with probability 0, and invest in Stock C with probability 0.

10)

11)

12) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. The experts cannot predict inflation for the next year. The company has three plans of action and will implement these at varying percentages of its total operation. The payoff matrix for these three plans is given below, with values given in hundred thousands. What is the marketing strategy for the company that will yield the best expected value?

12)

Above 6% 3-6% Below 3% Plan 1 8 16 15 Plan 2 8 9 10 Plan 3 17 12 14 A) The company should use Plan 1 with probability 1, Plan 2 with probability 0, and Plan 3 with probability 0. B) The company should use Plan 1 with probability 0, Plan 2 with probability 1, and Plan 3 with probability 0. C) The company should use Plan 1 with probability 5/13, Plan 2 with probability 0, and Plan 3 with probability 8/13. D) The company should use Plan 1 with probability 1/3, Plan 2 with probability 1/3, and Plan 3 with probability 1/3. Provide an appropriate response. 13) Is the following matrix game strictly determined? Answer Yes or No -1 8 -5 6 12 -2 4 -3 A) Yes B) No 14) Find the expected value of the matrix game M = 0 1 2 3 2 1 strategies: P = 3 3 and Q = 1 .

13)

1 for the respective row and column -1

14)

3 A)

2 9

1 3

4 9

Determine if the statement is true or false. 15) If a payoff matrix has a row consisting of all 0's, then that row is recessive. A) True B) False

D) 1

15)

Provide an appropriate response. 16) Find the saddle value, if it exists, for the matrix game (include row and column location). 3 2 1 -2 2 -2 3 1 -2 -2 1 -3 A) 3; row 3 column 1 C) 1; row 3 column 2

16)

B) 1; row 1 column 3 D) 3; row 1 column 1

17) In a two-finger Morra game, if player R matches player C, then R wins and if R and C don't match, then C wins. R will win $3 with one finger or two fingers, while C will win $1 with one finger and $5 with 2 fingers. The payoff matrix is 1 finger 2 fingers 1 finger 3 -5 2 fingers -1 3

17)

Find the optimal strategies P * and Q * for R and C respectively, and the value v of the game. 1 2 3 3 1 1 A) P * = 3 3 ; Q * = 1 ; v = 2 B) P * = [1 2]; Q * = 1 ; v = 2 3 3 3 2 3

2 1 3 ; Q* = 2 ; v = 3 3

2 C) P * = 3

1 D) P * = 3

2 3

2 1 3 ; Q* = 1 ; v = 3 3

18) A small town has two competing grocery stores, store R and store C. Each week each store decides to advertise its specials using either a newspaper advertisement or a mailing. The following payoff matrix indicates the percentage of market gain or loss for each choice of action by store R and store C. C R Paper Mail

Paper 2 -1

Mail -4 -3

Use linear programming and a geometric approach to find optimal strategies P * and Q * for store R and store C and the value v of the game. 3 7 10 10 3 3 2 2 A) P * = 5 5 ; Q * = 7 ; v = 1 B) P * = 5 5 ; Q * = 3 ; v = 1 5 5 10 10

C) P * =

7 10 3 10

2 ; Q* = 5

3 1 5 ;v= 5

3 D) P * = 5

2 5 ; Q* =

7 10

1 3 ;v= 5 10

18)

Use the simplex method to find the optimum strategy for players A and B and the value of the game for the payoff matrix. 19) 1 2 3 19) 1 2 3 -2 2 -3 0 -1 1 1 1, 2 1, 5 1, 2 A) P* = , B) P* = C) P* = D) P* = 2 2 3 3 6 6 3 3 1 3

1 6

1 3

2 3

Q* = 0 2 3

Q* = 0 5 6

Q* = 0 2 3

Q* = 0 1 3

value = - 4 3

value = - 3 2

value = -1

Provide an appropriate response. 20) The matrix for a strictly determined matrix game is given below. Determine if the game is fair or not fair. 5 -2 -4 -3 A) Fair

B) Not fair

21) Suppose a matrix game has the following nonstrictly determined matrix: M= 7 5 2 17 Set up (but do not solve), the two corresponding linear programming problem used to solve this matrix game. A) Minimize y = x1+ x2 B) Maximize y = x1+ x2 subject to

Maximize

20)

subject to 7x1+ 2x2 ≥ 1 5x1+ 17x2 ≥ 1

7x1+ 2x2 ≥ 1 5x1+ 17x2 ≥ 1

x1, x2 ≥ 0 y = z1+ z2

Minimize

subject to

subject to 7z1+ 5z2 ≤ 1

7z1+ 5z2 ≤ 1

2z1+ 17z2 ≤ 1

z1, z2 ≥ 0

21)

22) Find the expected value of the game matrix A = 4 0 if player 1 and player 2 decide on 2 3 1 2 1 3 strategies P = 4 4 and Q = 1 .

22)

2 A)

8 19

5 4

19 8

9 4

Solve the problem. 23) A company has three products that fluctuate in favorability. The payoff matrix for the expected profit in tens of thousands of dollars is given below. What are the percentages of each product that the company should make that would maximize the expected value of the profit? Round to the nearest tenth of a percent, if necessary.

23)

Preference Product 1 Product 2 Product 3 Product 1 7 15 14 Product 2 7 8 9 Product 3 16 11 13 A) Product #1: 0%, Product #2: 100%, Product #3: 0% B) Product #1: 100%, Product #2: 0%, Product #3: 0% C) Product #1: 33.3%, Product #2: 33.3%, Product #3: 33.3% D) Product #1: 38.5%, Product #2: 0%, Product #3: 61.5% Provide an appropriate response. 24) Delete recessive rows and columns from the following matrix game: 4 -3 4 -4 -1 2 0 0

24)

1 -2 0 -2 3 2 5 2

A) 4 -3 4 -4

-3 1 B) -4 0 2 3

4 -3 C) -1 2 0 0

D) 3 2 5 2

25) The U.S. Marines is playing a war game with side A trying to capture side B's headquarters. Side A has three strategies and side B has four defenses. The payoff matrix shows side A's percentage chance of winning. What is the value of the game? B -6 51 71 -21 A 51 -15 9 -41 76 28 -4 -75 A) -41 B) 51 C) -75 D) -21

25)

26) Is the following matrix game strictly determined? Answer Yes or No. 2 0 -1 3 6 0 1 3 7 A) Yes B) No

26)

27) Delete recessive rows and columns from the following matrix game: 4 -1 3 5 4 0 3 6 3 -3 1 4 5 -1 5 5 1 3 -1 1 0 3 A) 1 5 3 1

-3 1 B) -1 5 3 -1

5 5 -1 1

27)

0 3 D) -1 5 3 -1

28) Find the saddle values, if it exists, for the matrix game (include row and column location). 4 6 3 -6 A) 4; row 1 column 1 B) 6; row 1 column 2 C) 4; row 2 column 1 D) 3; row 2 column 1 Find the saddle point and value of the game. 29) Two merchants in the same city plan on selling a new product. Each merchant has 3 strategies to enhance sales. The strategies chosen by each will determine the percentage of sales of the product each gets. B -53 -10 -50 A 41 12 58 -50 -24 50 A) (2, 2), value 12 B) (3, 1), value -51 C) (1, 1), value -53 D) (1, 2), value -10 Provide an appropriate response. 30) Find the saddle values, if it exists, for the matrix game (include row and column location). 3 -5 2 6 -4 -3 2 -2 1 A) -4; row 2 column 2 B) -2; row 3 column 2 C) 6; row 2 column 1 D) -5; row 1 column 2 31) Solve the matrix game using a geometric linear programming approach. M = 4 -6 -2 3 3 3 5 5 1 2 2 1 * * * * A) P = 3 3 , Q = 2 , v = 0 B) P = 3 3 , Q = 2 , v = 0 5 5 2 5

1 C) P * = 3 , Q * = 3 5

2 3 , v= 0

1 D) P * = 3

2 5 2 * , Q = 3 3 ,v=0 5

28)

29)

30)

31)

32) Two supermarkets, R and C, want to run a promotional on the same item. A market research firm provided the payoff matrix below, where each entry indicates the percentage of customers going to market R at the indicated prices of that item. Find the saddle value, and the optimum strategy for each store. Store C $1.55 $1.85 $1.55 60% 50% Store R $1.85 55% 45% A) 60%; R sells the item at $1.85, C sells the item at $1.55. B) 60%; R sells the item at $1.55, C sells the item at $1.85. C) 50%; R sells the item at $1.55, C sells the item at $1.85. D) 50%; R sells the item at $1.85, C sells the item at $1.55.

32)

33) Find the expected value of the matrix game M = -1 0 for the respective row and column 2 1 1 3 1 1 strategies: P = 4 3 and Q = 2 .

33)

3 A)

2 9

4 9

C) - 1 12

13 36

Determine if the statement is true or false. 34) If the value of a matrix game is positive, then all payoffs are positive. A) False B) True Provide an appropriate response. 35) What is the optimal strategy for each player in the following matrix game? Use R for row and C for column. -1 1 1 5 4 -1 1 4 -1 5 -3 -1 5 -3 2 A) R plays row 2 or row 3, C plays column 1 B) R plays row 1 , C plays column 2 C) R plays row 1 or row 2, C plays column 1 D) R plays row 2, C plays column 4 Find the saddle point and value of the game. 36) Two countries are involved in a border war. Each country has 3 strategies with payoffs in square miles of land. Positive numbers represent gains by A. B 3 -3 -8 A 7 -4 -12 -10 5 -10 A) (3, 2), value 5 B) (3, 3), value -10 C) (1, 3), value -8 D) (1, 3), value -10

34)

35)

36)

Provide an appropriate response. 37) Is the following matrix game strictly determined? Answer Yes or No. 1 0 3 -1 2 1 2 2 3 A) Yes B) No

37)

38) Suppose a matrix game has the following nonstrictly determined matrix: M = 4 -6 -2 3 Set up (but do not solve), the two corresponding linear programming problem used to solve this matrix game. A) Maximize y = x1 + x2 B) Minimize y = x1 + x2 subject to

Minimize

38)

subject to 11x1 + 5x2 ≥ 1 x1 + 10x2 ≥ 1

11x1 + 5x2 ≥ 1 x1 + 10x2 ≥ 1

x1, x2 ≥ 0 y = z1+ z2

Maximize

subject to

subject to 11z1 + z 2 ≤ 1 5z1 + 10z2 ≤ 1

11z1 + z 2 ≤ 1 5z1 + 10z2 ≤ 1

z1, z2 ≥ 0

39) Find the saddle values, if it exists, for the matrix game (include row and column location). 8 1 2 5 6 2 A) 2; row 3 column 2 B) 2; row 2 column 1 C) 1; row 1 column 2 D) Does not exist

39)

40) The matrix for a strictly determined matrix game is given below. Determine if the game is fair or not fair.

40)

3 -4 -1 7 0 2 A) Fair

B) Not fair

Determine if the statement is true or false. 41) If M is a matrix game, then there is a fair matrix game in which the optimal strategies are those of M. A) True B) False

41)

Find the smallest integer k ≥ 0 such that adding k to each entry of the given matrix produces a matrix with all positive payoffs. 42) -2 4 42) 10 A) 1 B) 4 C) 3 D) 2

Use the simplex method to find the optimum strategy for players A and B and the value of the game for the payoff matrix. 43) 1 2 3 43) 1 -7 1 0 2 -10 -8 -7 3 2 -3 -1 2 , 0, 1 1 , 0, 2 A) P* = B) P* = 3 13 3 13 8 17 Q* =

8 17

9 17

Q* =

0 value = 5 72

247 value = 72 C) P* =

1 , 0, 1 2 2

D) P* =

8 17 Q* =

9 17

5 , 0, 8 13 13 4 13

Q* =

9 17

0 value = 5 72

9 13 0

value = -

19 13

44)

1 2 3 1 3 7 -4 2 1 -2 0 3 6 8 -1

44)

A) P* = 0,

Q* =

3 , 8 11 11

B) P* = 0,

0 1 11

Q* =

10 11

value = 149 121 C) P* = 0,

Q* =

9 , 2 11 11

0 3 11 8 11

value = 725 121

2, 1 3 3

D) P* = 0,

0 3 4

Q* =

1 4

value = 15 4

9 , 2 11 11

0 1 11 10 11

value = - 2 11

Solve the problem. 45) A company has three different marketing strategies that produce different results depending upon whether inflation is above 6%, between 3% and 6% inclusive, or below 3% annually. The experts cannot predict inflation for the next year. The company has three plans of action and will implement these at varying percentages of its total operation. The payoff matrix for these three plans is given below, with values given in hundred thousands. What is the market strategy for the company that will yield the best expected value? Above 6% 3-6% Below 3% Plan 1 9 17 16 Plan 2 18 13 15 Plan 3 9 10 11 A) The company should use Plan 1 with probability 5/13, Plan 2 with probability 8/13, and Plan 3 with probability 0. B) The company should use Plan 1 with probability 0, Plan 2 with probability 1, and Plan 3 with probability 0. C) The company should use Plan 1 with probability 4/13, Plan 2 with probability 9/13, and Plan 3 with probability 0. D) The company should use Plan 1 with probability 1, Plan 2 with probability 0, and Plan 3 with probability 0.

45)

Find the value of the game. 46) 1 2 3 1 -10 -2 -3 2 -1 -6 -4

A) Row player:

46)

1 2

1 2 ; Column player:

8 17

5 9 ; v = 72 17 0

B) Row player:

2 3

2 3 ; Column player:

8 17

377 9 ; v = - 36 17 0

C) Row player:

5 13

8 13 ; Column player:

4 13

58 9 ; v = - 13 13 0

D) Row player:

2 3

1 3 ; Column player:

8 17

5 9 ; v = 72 17 0

Find the saddle point and value of the game. 47) Suppose a rugby team with the ball (team A) can choose from three plays while the opposing team (B) has four possible defenses. The numbers in the payoff matrix represent yards gained by A. B 15 -3 -4 8 A 6 18 5 6 -8 -5 -2 6 A) (3, 4), value 6 B) (2, 2), value 18 C) (2, 3), value 5 D) (1, 1), value 15

47)

Provide an appropriate response. 48) Solve the matrix game M = 0 1 indicating optimal strategies P * and Q * for R and C 1 -1 respectively, and the value v of the game. 2 2 3 3 1 1 A) P * = 3 3 ; Q * = 1 ; v = 2 B) P * = [2 1]; Q * = 1 ; v = 1 3 3 3 2 C) P * = 3

1 3

1 1 3 ; Q* = 1 ; v = 3 3

48)

2 3

1 1 3 ; Q* = 1 ; v = 3 3

2 D) P * = 3

Use the simplex method to find the optimum strategy for players A and B and the value of the game for the payoff matrix. 49) 1 2 3 49) 1 -6 2 1 2 3 -2 0 2, 2 2, 1 A) P* = B) P* = 3 3 3 3 8 17 Q* =

8 17

9 17

Q* =

0 value = 5 72

13 value = 12 C) P* =

1, 1 2 2

D) P* =

8 17 Q* =

9 17

5 , 8 13 13 4 13

Q* =

9 17

0 value = 5 72

9 13 0

value = -

6 13

Provide an appropriate response. 50) Two Wally-Worlds are going to build within a three block area. The exact sites will determine the percentage of business above or below 50% each gets, as shown in the payoff matrix. What is the value of the game? B 12 8 11 A 3 -3 6 2 2 -8 A) 6 B) 12 C) 11 D) 8

50)

51) Two racquetball clubs are in competition in attracting new members. Advertising campaigns are being contemplated by both clubs which emphasize either beginning, intermediate, or advanced player tournaments among members. After careful research, the following results were found:

51)

Racketball Club B Beginning Intermediate Advanced Beginning 25% 85% 15% Racketball Club A Intermediate 45% 65% 75% Advanced 35% 25% 55% Thus, if beginner tournaments are featured by both clubs, Club A expects to attract 25% of the new members and Club B expects to attract 75% of the new members, etc. Find the value of the game. A) 85% B) 65% C) 45% D) 55% 52) QMC decided to put its new product on the market with ads in a trade magazine and a booth at a trade show. It found out that its major competitor ZMC also had decided to advertise the same way for its new product. The payoff matrix shows the increased sales for QMC, as well as decreased sales for ZMC. What is the optimum strategy for QMC and the value of the game? 1.0 -0.7 -0.5 0.5 A) QMC should use ads with probability 10 and use booths with probability 17 . The value 27 27 of the game is 1 . 18 B) QMC should use ads with probability 17 and use booths with probability 10 . The value 27 27 of the game is 1 . 27

52)

Answer Key Testname: CHAP 11_14E

1) B 2) A 3) A 4) A 5) A 6) A 7) A 8) B 9) B 10) A 11) B 12) C 13) B 14) B 15) B 16) B 17) D 18) B 19) B 20) B 21) A 22) C 23) D 24) C 25) D 26) B 27) D 28) A 29) A 30) B 31) A 32) C 33) D 34) A 35) C 36) C 37) A 38) B 39) D 40) A 41) A 42) C 43) D 44) D 45) A 46) C 47) C 48) D 49) D 16

Answer Key Testname: CHAP 11_14E

50) D 51) C 52) A

CHAPTER 12: APPEX A

Exam Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Which of the following sets: N, Z, Q and I, together compose the set of real numbers? A) I and Q together B) I and Z together C) Z and Q together D) N and Z together

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) Simplify and express using positive exponents:

3) Write in simplest radical form:

-1 4x2/3 10x-1/4

243x13 y 22

4) The supply and demand equations for a certain product are s = 2,500p - 14,500 and d = 3,000 , where p is the price in dollars. Find the price where supply equals demand. p Simplify the expression by combining like terms, if possible. 5) Perform the indicated operation and simplify: 3t - [5 - 3{t - t(6 + t)}].

Provide an appropriate response. -4 6) Simplify and express the answer using positive exponents: 5m m-8

7) Factor, if possible, as the product of two first-degree polynomials with integer coefficients: 4x2 - 131x + 435

8) Each ounce of food A contains 4 units of calcium, and each ounce of food B contains 7 units of calcium. A 150 ounce diet mix is formed using foods A and B. If x is the number of ounces of food A used, write an algebraic expression that represents the total number of units of calcium in the diet mix. Simplify the expression.

9) Perform the indicated operation and reduce to lowest terms: 3x + 3y 6x + 6y ÷ x2 - 5xy - 14y 2 x2 + 9xy + 14y 2

10) Indicate True (T) or False (F), and for each false statement find real number replacements for a, b, and c that will provide a counterexample. (A) a - b = b - a (B) (a - b) - c = a - (b - c) (C) a + b = b + a (D) (a ∙ b) ∙ c = a ∙ (b ∙ c)

11) Express as a simple fraction reduced to lowest terms:

a - 1 - 2b b a a - 3 + 2b b a

10)

11)

12) Convert the repeating decimal 0.454 545 . . . into a reduced fraction.

12)

13) Factor by grouping: 12x3 - 8x2 + 3x - 2

13)

14) Write 7.214 × 10 7 in standard notation.

14)

3 3 2 4 15) Write 3x (x -2) - 2x (x - 2) with positive exponents only, and as a single fraction (x - 2) 2

15)

reduced to lowest terms. 16) Which polynomial can be factored using integer coefficients? Find its factored form. I) 20x2 - 3x - 35

16)

II) 16m2 + 25n2 III) x2 + 7x - 5 3 2 17) Perform the indicated operation and reduce to lowest terms: 20x y ∙ x - 4 4xy + 8y 16x2 - 32x

17)

12m8 n6

18)

18) Change to simplest radical form:

3m8 n2

19) Change to simplest radical form: (2xy6 z35 ) 3/7

19)

5 3 20) Write 3x - 4x + 9x in the form axp + bx q + cxr, where a, b, and c are real numbers and 3x2

20)

p, q and r are integers. 21) According to the 2000 U.S. census, the population of the United States on April 1, 2000 was approximately 281,422,000 (Source: www.census.gov). The population of the U.S. in 1900 was approximately 76,212,000. Write both population numbers in scientific notation and use these expressions to calculate the ratio of the population in 2000 to that in 1900. Express the ratio in standard decimal form to four decimal places. 2

21)

22) Simplify and express the answer using positive exponents only: (2x-5/8 y 3/4) 8

22)

23) Indicate True (T) or False (F) by each statement. (A) Every real number is a natural number. (B) An irrational number is not a real number. (C) Every natural number is an integer. (D) An integer is a rational number.

23)

24) Solve by factoring: 4y 2 = 10y

24)

25) Combine into a single fraction and simplify:

y-5 y +5 - 3 2 2 y - 4 y - 4y + 4 2 - y

25)

26) Factor the expression using integer coefficients: 24a2x3 - 3a2y 3 .

26)

27) Given the sets N, Z, Q, and R, indicate to which set(s) each of the following numbers belong. (A) 5 6

27)

(B) e (C) -2 (D) 8 28) Factor using integer coefficients: 4x3 y 2 - 20x3 y + 25x3

29) Simplify and express using positive exponents:

28)

5m-3 n4

-2 29)

4 30) Change to simplest radical form: 49x y 7x

30)

31) Convert the repeating decimal 2.626 262 . . . into a reduced fraction.

31)

32) Combine into a single fraction and simplify:

1 49x4

- 7x + 1 + 5 245x 5x3

32)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify. The exponents in the answer should be positive integers. -7ym3 n 33) -4ym6 n A)

3y 2n2 m3

4m3

33) 1

3m3

3 m3

Simplify the expression. 4x + 2 34) 2 20x + 18 x + 4

34)

1 5x + 2

4x + 5 5x + 18

4x 5x + 2

4x + 2 2 20x + 18 x + 4

Rewrite the expression with a positive rational exponent. Simplify, if possible. 35)

9a2b -4 1/2 a-2b 4

35)

A) 3a2b 4

3a2 b4

9a b2

9b 4 a

Use the quadratic formula to solve the equation. 36) x2 + 18 x + 71 = 0 A) {9 C) {9 +

71, 9 + 10}

36)

71}

B) {-9 - 10, -9 + D) {-18 + 71}

10}

Simplify the expression by combining like terms, if possible. 37) -3(10r + 2) + 5 (2r + 3 ) A) -20r + 2 B) 7r - 1 C) -20r + 9

37) D) -36r

Find the product. 38) (6y + 11)(5y 2 - 2y - 10) A) 30y 3 - 12y 2 - 60y + 11

B) 30y 3 + 67y 2 + 82y + 110

C) 85y 2 - 34y - 170

D) 30y 3 + 43y 2 - 82y - 110

38)

Rationalize the denominator and simplify. Assume that all variables represent positive real numbers. 5 39) 2+ 7 A)

Use the square root property to solve the equation. 40) (x + 3 ) 2 = 12

39)

40)

A) {-3 - 2 3, -3 + 2 3} C) {2 3 - 3, 2 3 + 3}

B) {-3 - 2 6, -3 + 2 D) {-2 3, 2 3}

Find the product. 41) (3x + 12y)(5x + 8 y) A) 15x2 + 84xy + 84y 2

41) B) 15x2 + 60 xy + 96 y 2 D) 15x2 + 84xy + 96 y 2

C) 15x2 + 24xy + 96 y 2

Factor the GCF from the polynomial. 42) 15x6 y 6 z - 25x5 y 5

42)

A) x5 y 5 (15xyz - 25) C) 5x5 y 5 z(3xy - 5)

B) 5x5 y 5 (3xyz - 5) D) 5xy(3x5 y 5 z - 5 x4y 4)

Factor completely using grouping. 43) 18y 2 + 81y - 45

43)

A) (18y - 9 )(y + 5) C) 9(2y + 1)(y - 5)

B) 9(2y - 1)(y + 5) D) prime

Simplify. The exponents in the answer should be positive integers. 44) (7x-3 y 3 )(7-1x7y -8 ) 4 A) - x 7y 5

1 x21y 24

x10 y 11

44) D)

x4 y5

Answer Key Testname: CHAP 12_14E

1) A 2)

5 11/12 2x

3) 3x2y 4 4) $6

x3 y 2

5) -5 - 12t + 3t2 6) 5m4 7) (4x - 15)(x - 29) 8) 1050 -3x x + 7y 9) 2(x - 7y) 10) (A) F, a = 1 and b = 2 provides a counterexample (B) F, a = 3, b = 2, and c = 1 provides a counterexample (C) T (D) T a +b 11) a-b 12)

5 11

13) (3x - 2)(4x2 + 1) 14) 72,140,000 15) x2(x - 2)(x + 4) 16) I; (4x + 5)(5x - 7) 5x2 17) 16 18) 6m8 n4 7 19) y 2z15 8x3 y 4 20) x3 - 4 x + 3x-1 3 21) 2.81422 × 10 8 , 7.6212 × 10 7; 3.6926 256y 6 22) x5 23) (A) F; (B) F; (C) T; (D) T 24) y = 0 or 5 2 25)

3y 2 - 14y - 12 (y + 2) (y - 2) 2

26) 3a2(2x - y)(4x2 + 2xy + y 2) 27) (A) Q, R; (B) R; (C) Z, Q, R; (D) N, Z, Q, R 28) x3 (2y - 5) 2 6

Answer Key Testname: CHAP 12_14E

29)

m6 n8 25

30) 7x3 y 260 31) 99 32)

5 - 49x - 343x2 + 5x3 245x4

33) B 34) A 35) B 36) B 37) C 38) D 39) B 40) A 41) D 42) B 43) B 44) D

CHAPTER 13: APPEX B

Exam Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Write the first four terms of the sequence an = n[9 + 8(-1) n].

2) Indicate by letter which of the following sequences can be the first three terms of a geometric sequence and state the common ratio for those that are. (A) 1, -4, 16, . . . (B) 14, 2, 2 , . . . (C) 1, -8, -64, . . . 7

3) Evaluate: C40, 37

4) If a person borrows $13,200 and agrees to repay the loan by paying $200 per month to reduce the loan and 1% of the unpaid balance each month for using the money, what is the total cost of the loan over 66 months?

5) Find the general term of a sequence whose first four terms are 3 , 6 , 9 , 12 . 5 6 7 8

6) Indicate by letter which of the following sequences can be the first three terms of an arithmetic sequence and state the common difference for those that are. (A) 9, 3, -3, . . . (B) 2, 6, 10, . . . (C) 5, 8, 12, . . .

7) Find the sum of the first 25 terms of the geometric sequence 250, 250(1.05), 250(1.05) 2, . . .

8) Evaluate: 46! 38!8!

9) Find the 67th term of the sequence defined by an = n + 3 . n-1

10) Find the sum of all the odd integers between 52 and 346.

10)

11) Find the 300th term and the sum of the first 300 terms for the arithmetic sequence 8, 11, 14, . . .

11)

12) Write the following sum using summation notation: 1 - 2 a + 3 a2 - 4 a3 . . . + 15 a14 2 3 4 5 16

12)

13) Find the sum of the infinite geometric sequence (if it exists): 4, - 8 , 16 , - 32 , . . . 3 9 27

13)

14) Find the sixth term in the expansion of (p - 2q) 12.

14)

15) Find the common ratio of a geometric sequence if the first term is 5 and the 12th term is 30.

15)

16) Find the first five terms of the sequence defined by the recursive formula a1 = 2, an =

16)

4an - 1 - 1 for n ≥ 2. 17) Find the sum of the infinite geometric sequence (if it exists): 7, 7 , 7 , . . . 5 25

17)

18) Write the alternating series - 1 + 1 - 1 + 1 - 1 using summation notation with the 2 3 4 5 6

18)

summing index k starting at k = 1. 4 19) Write ∑ k=1

k without summation notation. Do not evaluate. k + 13

20) Expand: (3x + y) 4

19)

20)

Answer Key Testname: CHAP 13_14E

1) 1, 34, 3, 68 2) (A) Common ratio = -4 (B) Common ratio = 1 7 3) 9,880 4) $4,422 3n 5) n +4 6) (A) Common difference = -6 7) 11,931.77 8) 260,932,815 35 9) 33

(B) Common difference = 4

10) 29,253 11) a300 = 905, s 300 = 136,950 12)

∑ (-1) n + 1 n n+ 1 an - 1

k=1 12 = 2.4 13) 5

14) -25,344p 7q 5 15) 1.18 16) 2, 7, 27, 107, 427 35 = 8.75 17) 4 18)

∑ (-1) k +1

k=1 1 + 2 + 3 + 4 19) 14 15 16 17 20) 81x4 + 108x3 y +54x2y 2 + 12xy3 + y 4