It is possible to model the flight of a tennis ball that has just been served down the center of the court by the equation

where y is the height (in feet) of a tennis ball that is being served at an angle α relative to the horizontal axis, x is the horizontal distance (in feet) that the ball has traveled, h is the height of the ball when it leaves the server’s racket and V is the velocity of the tennis ball when it leaves the server’s racket.

(a) If a tennis ball is served from a height of 9 ft and the net is 3 ft high and 39 ft away from the server, does the tennis ball that is hit with a velocity of 50 mph (approximately 73 ft per second) make it over the net if it is served at an angle of π/24?

(b) When y = 0, the corresponding value of x gives the total distance that the tennis ball has traveled while in flight (provided that it cleared the net). For a serving height of 9 ft, the equation for calculating the distance traveled is given by

Use the TABLE function on a graphing calculator or a spreadsheet to determine a range of angles for which the tennis ball will clear the net and travel between 39 and 60 ft when it is hit with an initial velocity of 44 ft per second.

(c) Because calculating dx/dα is so complicated analytically, use a graphing calculator to estimate this derivative when the initial velocity is 44 ft per second and α = π/8. Interpret your answer.

Transcribed Image Text:

y = x tan a - 16x² V² -sec² a + h,