Junior Philippine Institute of Chemical Engineers- Xavier University Department of Academics MODULE 1: PRE-TEST SOLUTIONS General Engineering Principles
John’s father is 5 times older than John and John is twice as old as his sister Alice. In two years time, the sum of their ages will be 58. How old is John now? Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours to paint a similarlysized house. How long would it take the two painters together to paint the house? The average of 10 student scores is 82, but the student mistakenly recorded two values of 79 as 78 and three values of 83 as 85. What is the true average of the set of scores? Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What are the numbers? Find the domain of 2đ?‘Ľâˆ’8 đ?‘Ľ 2 −16
IN 2 YEARS x+2 5x + 2 0.5x + 2 58
(x+2)+(5x+2)+(0.5x+2) = 58 Solving for x, x = 8
Painter 1 Painter 2
TIME per unit job 12 hr/job 8 hr/job
Total rate =
1 đ?‘—đ?‘œđ?‘? 12 â„Žđ?‘&#x;
+
1 đ?‘—đ?‘œđ?‘? 8 â„Žđ?‘&#x;
JOB per unit time 1/12 job/hr 1/8 job/hr = 0.2083
(-8 – h)2 + (-6 – k)2 = 102 64 + 16h + h2 + 36 + 12k + k2 = 100 16h + 12k = 0
Find the major axis of an ellipse with equation 12x2 + 24y2 - 48x + 144y + 216 = 0.
đ?‘—đ?‘œđ?‘? â„Žđ?‘&#x;
Time for single job by all units = (0.2083 đ?‘—đ?‘œđ?‘? -1 ) = 4.8 hours/job â„Žđ?‘&#x;
Total score = (10 students)(82) = 820 Adjustments
= 2(79-78) = +2 = 3(83-85) = -6
True actual score = 820 + 2 – 6 = 816
What is the slope of a line perpendicular to a second line passing through (5,2) and (9,1)? 3đ?‘Ľâˆ’5 For đ?‘“(đ?‘Ľ) = , 4đ?‘Ľ find đ?‘“ −1 (2)?
True average = 816/10 = 81.6
Let x be the smaller number Let y be the larger number 2y = 3 + 5x 4y + 3x =71
Find the vertical asymptotes of đ?‘Ś = 2đ?‘Ľ . 2 đ?‘Ľ −3đ?‘Ľâˆ’10
The value of 0.25π rad in degrees is: The exact value of sin(15°) is:
Solving simultaneously, use [MODE] – [5] – [1], x, y = {5,14}
(x + 4)(x – 4)
(x + 4)
Finding the zeros of the denominator, (x + 4) = 0 → x = –4 General equation of circle: (x – h)2 + (y – k)2 = r2 Substituting, (6 – h)2 + (8 – k)2 = 102 36 – 12h + h2 + 64 -16k + k2 = 100 -12h – 16k = 0
Solving, (h, k) = (0, 0) General equation of ellipse: (đ?‘Ľâ€“ â„Ž)2 (đ?‘Śâ€“ đ?‘˜)2 + = đ?‘&#x;2 2 đ?‘Ž đ?‘?2 Completing the square of given equation, đ?&#x;? 12x2 + 24y2 - 48x + 144y + 216 = 0 ( ) đ?&#x;?đ?&#x;? 2 2 x + 2y - 4x + 12y + 18 = 0 [(x2 - 4x)+4] + [2(y2 + 6y)+9] = -18+4+18 2 (x – 4x + 4) + 2(y2 + 6y + 9) = 4 (đ?‘Ľâ€“ 2)2 (đ?‘Ś + 3)2 + = 1 4 2 Since a2 = 4 → a = 2 Major axis = 2a = 2(2) → 4 Δđ?‘Ś 1−2 1 Slope of points: → =− Δđ?‘Ľ
9−5
Perpendicularity → m’ = − m’ = −
1 1 4
(− )
4
1
3đ?‘Ľâˆ’5
= The ordinate value for θ = 60° is:
√2 √3 2 2
−
√2 1 2 2
=
√6 4
−
√2 4
Find the sum of the integers from 1 to 100.
√đ?&#x;”−√đ?&#x;?
cos đ?‘Ľ sin đ?‘Ľ
Solving simultaneously, a0 = 6 and d = 4 a100 = 6 + 99(4) = 402 By direct input into calculator, 100
∑ đ?‘– = 5050 đ?‘–=1
Evaluate đ?‘Ľ = 3 log 5 √25
√3
+
2
Find the sum of the geometric series 1 1 1 {1, , , ‌ }. 4 16 64
đ?&#x;’
đ??…
From graph, range → [đ?&#x;Ž, ) âˆŞ ( , đ??…] đ?&#x;? đ?&#x;? General sequence term: an = a0 + (n-1)d Term expressions: a5 = a0 + (5-1)d a15 = a0 + (15-1)d 22 = a0 + 4d 62 = a0 + 14d
Alternatively, 1 100 2 99 50 3 97 pairs of 101 4 96 â‹Ž â‹Ž By direct input into calculator, 3 đ?‘Ľ = log 5 √25 2 Solving for x → đ?‘Ľ = , 3 By principle, 3 đ?‘Ľ = log 5 √25 3
Ordinate → y (Abscissa → x) For points in unit circle, (x,y) → (cos θ, sin θ)
cot đ?‘Ľ + tan đ?‘Ľ =
An arithmetic sequence has its 5th term equal to 22 and its 15th term equal to 62. Find its 100th term.
4
Ordinate = sin θ = sin 60° = The identity (cot � + tan �) is equivalent to:
đ??…
√6−√2
→
cos 2 đ?‘Ľ + sin2 đ?‘Ľ 1 = sin đ?‘Ľ cos đ?‘Ľ sin đ?‘Ľ cos đ?‘Ľ = đ??œđ??Źđ??œ đ??ą đ??Źđ??žđ??œ đ??ą
The range of the inverse secant function is:
=4
đ?‘Ś= → 4đ?‘Ľđ?‘Ś = 3đ?‘Ľ − 5 4đ?‘Ľ 4đ?‘Ľđ?‘Ś − 3đ?‘Ľ = −5 → đ?‘Ľ(4đ?‘Ś − 3) = −5 −5 −đ?&#x;“ đ?‘Ľ= → đ?’šâˆ’đ?&#x;? = 4đ?‘Śâˆ’3 đ?&#x;’đ?’™âˆ’đ?&#x;‘ −5 −5 y −1 (2) = = = −đ?&#x;? 4(2) − 3 5 Vertical asymptotes are the roots of the denominator: (đ?‘Ľ 2 − 3đ?‘Ľ − 10) = 0 → (x – 5)(x + 2) = 0 V.A. → x = 5, x = -2 180° 0.25đ?œ‹ ∙ = đ?&#x;’đ?&#x;“° đ?œ‹ By direct input to scientific calculator, sin (15°) =
=
đ?‘š
By principle, sin (15°) = sin (45°-30°) = sin45° cos 30° – cos45° sin30°
Domain is by default all the values of x minus the ones that will make the denominator zero. 2đ?‘Ľâˆ’8 2(đ?‘Ľâˆ’4) 2 → → 2 đ?‘Ľ −16
Find the center of a circle passing through (6,8) and (8,-6) with a radius of 10.
NOW x 5x 0.5x
John Father Alice
3
2
đ?&#x;?
5đ?‘Ľ = √25 = √52 = 53 → đ?’™ = đ?&#x;‘ Converting the series into expressions of integers, 1 1 1 {1, , , ‌ } →{40 , 4−1 , 4−2 , 4−3 ‌ } 4 16 64 Thus, the series can be expressed as ∞
∑ 4−đ?‘– đ?‘–=0
= 0.866
sin đ?‘Ľ cos đ?‘Ľ
The infinite sum cannot be plugged into a calculator, but note that as i increases the actual value decreases. The sum will be obvious with an upper limit of ~10.
Junior Philippine Institute of Chemical Engineers- Xavier University Department of Academics MODULE 1: PRE-TEST SOLUTIONS General Engineering Principles 10
∑ 4−đ?‘– = đ?&#x;?. đ?&#x;‘đ?&#x;‘đ?&#x;‘ đ?‘–=0
A trench is 180 ft long and 12 ft deep, 7 ft wide at the top and 4 ft wide at the bottom. How many cubic yards of earth have been removed? For a cube measuring 3 units on an edge, how many times must it be cut completely through the cube to make cubes measuring 1 unit on an edge? If a cube has an edge equal to the diagonal of another cube, find the ratio of their volumes. A rectangular field is to be fenced with 500 feet of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the shorter dimension of the field that will enclose the largest area. A person 100 meters from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees. Estimate the height of the tree. An integer from 100 through 999 is to be chosen at random. What is the probability that
V = BL đ??ľ1+đ??ľ2 B (area of trapezoidal face) = đ??ť= 2
7đ?‘“đ?‘Ą+4đ?‘“đ?‘Ą
∙ 12 đ?‘“đ?‘Ą= 66 ft2 2 V = (180 ft)(66 ft2) = 11 880 ft3 Converting, 1 đ?‘Śđ?‘‘ 3 11 880 đ?‘“đ?‘Ą 3 ∙ ( ) = 1320 đ?‘Śđ?‘‘ 3 3 đ?‘“đ?‘Ą
Cube 1 x x3
Edge Volume
Ratio of volumes =
(√2x)3 x3
Cube 2 √2x (√2x)3 3
=(√2) = 2.83
L W
W ______ B UI LDI NG
L + 2W = 500 ft A = LW = W(500- 2W) = 500W – 2W2 đ?‘‘đ??´ = 500 – 4W = 0 đ?‘‘đ?‘Š 4W = 500 → W = 125 ft
Htree = 100tan(18°) = 32.5 m
0 in three digits: 0 0 in two digits: (100, 200‌900) → 9 0 in one digit: (101, 102‌ 180, 190) x 9 → 162
the chosen number will have at least one digit be zero? From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done? A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw? A large box moving across a floor at constant speed has two people moving it. One is pushing 236.1 N from behind while the other is pulling 89.3 N from the front. What is the force of friction? A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of 0.25 ft/sec. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing? A spot light is on the ground 20 ft away from a wall and a 6 ft tall person is walking towards the wall at
P=
0+9+162 999−99
=171/900
C = 73đ??ś ∙ 62đ??ś + 74đ??ś ∙ 61đ??ś + 75đ??ś ∙ 60đ??ś C = (35 x 15) + (35 x 6) + (21 x 1) C = 756
C = 31đ??ś ∙ 62đ??ś + 32đ??ś ∙ 61đ??ś + 33đ??ś ∙ 60đ??ś C = 64
Constant speed → a = 0
→ 236.1 N
→ 89.3 N
FTOT - Ff = ma = 0 FTOT = Ff Ff = 236.1 + 89.3 = 325.4 N Solution found in Module 2.
Since the person is walking towards the wall, the shadow is decreasing.
a rate of 2.5 ft/sec. Is the shadow increasing or decreasing in height at this time?