582 CHAPTER 5 Trigonometric Functions 88. (Modeling) Distance of a Shot Put A shot-putter trying to improve performance may wonder whether there is an optimal angle to aim for, or whether the velocity (speed) at which the ball is thrown is more important. The figure shows the path of a steel ball thrown by a shot-putter. The distance D depends on initial velocity v, height h, and angle u when the ball is released. D h u One model developed for this situation gives D as D= v2 sin u cos u + v cos u 21v sin u22 + 64h 32 . Typical ranges for the variables are v: 33–46 ft per sec; h: 6–8 ft; and u: 40°9 45°. (Data from Kreighbaum, E., and K. Barthels, Biomechanics, Allyn & Bacon.) (a) To see how angle u affects distance D, let v = 44 ft per sec and h = 7 ft. Calculate D, to the nearest hundredth, for u = 40°, 42°, and 45°. How does distance D change as u increases? (b) To see how velocity v affects distance D, let h = 7 and u = 42°. Calculate D, to the nearest hundredth, for v = 43, 44, and 45 ft per sec. How does distance D change as v increases? (c) Which affects distance D more, v or u? What should the shot-putter do to improve performance? 89. (Modeling) Highway Curves A basic highway curve connecting two straight sections of road may be circular. In the figure in the margin, the points P and S mark the beginning and end of the curve. Let Q be the point of intersection where the two straight sections of highway leading into the curve would meet if extended. The radius of the curve is R, and the central angle u denotes how many degrees the curve turns. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) (a) If R = 965 ft and u = 37°, find the distance d between P and Q. (b) Find an expression in terms of R and u for the distance d between M and N. 90. (Modeling) Stopping Distance on a Curve Refer to Exercise 89. When an automobile travels along a circular curve, objects like trees and buildings situated on the inside of the curve can obstruct the driver’s vision. In the figure, the minimum distance d that should be cleared on the inside of the highway is modeled by the equation d = R a1 - cos u 2b . (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) (a) It can be shown that if u is measured in degrees, then u ≈ 57.3S R , where S is the safe stopping distance for the given speed limit. Compute d to the nearest foot for a 55 mph speed limit if S = 336 ft and R = 600 ft. (b) Compute d to the nearest foot for a 65 mph speed limit given S = 485 ft and R = 600 ft. (c) How does the speed limit affect the amount of land that should be cleared on the inside of the curve? h U P S Q N M C R R 2 u 2 u d R R d u NOT TO SCALE
RkJQdWJsaXNoZXIy NjM5ODQ=