NOONTIME TEMPERATURES IN HILO, HAWAII 2. Mon Tue Wed Thu Fri Sat Sun 66 78 75 69 78 77 70 The table above shows the temperatures at noon, in degrees Fahrenheit, in a city in Hawaii over a oneweek period. If m represents the median of these temperatures, f represents the temperature that occurred most often, and a represents the average (arithmetic mean) of these seven temperatures, which of the following is the correct order of m, f, and a ? (A) a < m < f (B) a < f < m (D) m < f < a (C) m < a < f (E) a = m < f Correct answer: (A) / Difficulty level: Medium For 100,000 cartridges sold at $10 per cartridge, s = 100 (since s is the number of cartridges sold, in thousands) and p = 10. Substituting into the equation 3000 yields 100 = . Solving this equation for a yields 2 (10 ) + a 100 ( 20 + a ) = 3000 20 + a = 30 a = 10 Since a is a constant, the function can be written as 3000 . To determine how many cartridges will s ( p) = 2 p + 10 be sold at $20 per cartridge, you need to evaluate 3000 s ( 20 ) = = 60 . Since s is given in thousands, 2 ( 20 ) + 10 there will be 60,000 cartridges sold at $20 per cartridge. To determine the correct order of m, f, and a, it is helpful to first place the seven temperatures in ascending order as shown below: 66 69 70 75 77 78 (1, 2) 78 The median temperature is the middle temperature in the ordered list, which is 75, so m = 75. The temperature that occurred most often, or the mode, is 78, so f = 78. To determine the average, you can add the seven numbers together and divide by 7. However, you can determine the relationship between the average and the median by inspection. The three numbers greater than 75 are closer to 75 than are the three numbers smaller than 75. Therefore, the average of the seven numbers will be less than 75. The correct order of m, f, and a is a < m < f . 3. y The projected sales volume of a video game 3000 , cartridge is given by the function s ( p ) = 2p + a where s is the number of cartridges sold, in thousands; p is the price per cartridge, in dollars; and a is a constant. If according to the projections, 100,000 cartridges are sold at $10 per cartridge, how many cartridges will be sold at $20 per cartridge? (A) 20,000 (B) 50,000 (C) 60,000 (D) 150,000 (E) 200,000 Correct answer: (C) / Difficulty level: Medium O 4. x In the xy-coordinate plane above, line contains the points (0, 0) and (1, 2). If line m (not shown) contains the point (0, 0) and is perpendicular to , what is an equation of m ? 1 1 (B) y = − x + 1 (C) y = − x (A) y = − x 2 2 (D) y = − x + 2 (E) y = −2 x Correct Answer: (A) / Difficulty level: Medium Using the coordinates of the two points given on line , 2−0 = 2. Line m, which is perpendicular 1− 0 1 to , will have a slope of − , since slopes of perpendicular 2 the slope of is lines are negative reciprocals of each other. An equation 1 of m can be written as y = − x + b . Since line m also 2 contains point (0, 0), it follows that b = 0. Therefore, an 1 equation of line m is y = − x. 2 The Mathematics Section 21