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Chapter 3 | Derivatives
298. [T] A television camera at ground level is 2000 feet away from the launching pad of a space rocket that is set to take off vertically, as seen in the following figure. The angle of elevation of the camera can be found by θ = tan −1 ⎛ ⎝ x 2000 ⎞ ⎠ , where x is the height of the rocket. Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5000 feet apart.
299. [T] A local movie theater with a 30-foot-high screen that is 10 feet above a person’s eye level when seated has a viewing angle θ (in radians) given by θ =cot −1 x 40 −cot −1 x 10 , where x is the distance in feet away from the movie screen that the person is sitting, as shown in the following figure.
a. Find dθ dx . b. Evaluate dθ dx
for x =5, 10, 15, and 20.
c. Interpret the results in b.. d. Evaluate dθ dx for x = 25, 30, 35, and 40 e. Interpret the results in d. At what distance x should the person stand to maximize his or her viewing angle?
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