84 CHAPTER 2 Functions and Their Graphs x 1 A(x) 5 4x 1 2 x2 36. Effect of Elevation on Weight If an object weighs m pounds at sea level, then its weight W (in pounds) at a height of h miles above sea level is given approximately by W h m h 4000 4000 2 ( ) ( ) = + (a) If Amy weighs 120 pounds at sea level, how much will she weigh on Pikes Peak, which is 14,110 feet above sea level? (b) Use a graphing utility to graph the function W W h . ( ) = Use m 120 pounds. = (c) Create a TABLE with TblStart 0 = and Tbl 0.5 Δ = to see how the weight W varies as h changes from 0 to 5 miles. (d) At what height will Amy weigh 119.95 pounds? (e) Does your answer to part (d) seem reasonable? Explain. 37. Cost of Transatlantic Travel A Boeing 787 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by C x x x 100 10 36,000 ( ) = + + where x is the groundspeed airspeed wind . ( ) ± (a) What is the cost when the groundspeed is 480 miles per hour? 600 miles per hour? (b) Find the domain of C. (c) Use a graphing utility to graph the function C C x . ( ) = (d) Create a TABLE with TblStart 0 = and Tbl 50. Δ = (e) To the nearest 50 miles per hour, what groundspeed minimizes the cost per passenger? 38. Reading and Interpreting Graphs Let C be the function whose graph is given below. This graph represents the cost C of manufacturing q computers in a day. (a) Find C 0 . ( ) Interpret this value. (b) Find C 10 . ( ) Interpret this value. (c) Find C 50 . ( ) Interpret this value. (d) What is the domain of C? What does this domain imply in terms of daily production? (e) Describe the shape of the graph. (f) The point 30, 32 000 ( ) is called an inflection point. Describe the behavior of the graph around the inflection point. q C 10 20 30 40 50 Number of Computers Cost (dollars per day) 60 70 80 150 000 100 000 50 000 (0, 5000) (10,19 000) (30, 32 000) (50, 51 000) (80, 150 000) per second. Suppose a player shoots a ball with an initial velocity of 28 feet per second. (a) Find the height of the ball after it has traveled 8 feet in front of the foul line. (b) Find the height of the ball after it has traveled 12 feet in front of the foul line. (c) Find additional points and graph the path of the basketball. (d) The center of the hoop is located 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Why or why not? If not, with what initial velocity must the ball be shot in order for the ball to go through the hoop? Source: The Physics of Foul Shots, Discover, Vol. 21, No. 10, October 2000. 34. Cross-sectional Area The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function A x x x 4 1 , 2 ( ) = − where x represents the length, in feet, of half the base of the beam. See the figure. (a) Find the domain of A. (b) Use a graphing utility to graph the function A A x . ( ) = (c) Create a TABLE with TblStart 0 = and Tbl 0.1 Δ = for x 0 1. ≤ ≤ Which value of x maximizes the crosssectional area? What should be the length of the base of the beam to maximize the crosssectional area? 35. Motion of a Golf Ball A golf ball is hit with an initial velocity of 130 feet per second at an inclination of 45° to the horizontal. In physics, it is established that the height h of the golf ball is given by the function h x x x 32 130 2 2 ( ) = − + where x is the horizontal distance that the golf ball has traveled. (a) Determine the height of the golf ball after it has traveled 100 feet. (b) What is the height after it has traveled 300 feet? (c) What is h 500 ? ( ) Interpret this value. (d) How far was the golf ball hit? (e) Use a graphing utility to graph the function h h x . ( ) = (f) Use a graphing utility to determine the distance that the ball has traveled when the height of the ball is 90 feet. (g) Create a TABLE with TblStart 0 = and Tbl 25. Δ = To the nearest 25 feet, how far does the ball travel before it reaches a maximum height? What is the maximum height? (h) Adjust the value of Tbl Δ until you determine the distance, to within 1 foot, that the ball travels before it reaches its maximum height. Credit: Paul Burns/Photodisc/ Getty Images
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