Big Brother's Eye
Alex is walking down the street at 4 feet per second. He is walking away from a telephone pole topped with a rotating surveillance camera. This camera follows Alex as he walks. The camera, which can only rotate up or down, is 20 feet above street level. At what speed (in radians per second) is the camera rotating when Alex is 15 feet from the pole? (Out to three decimals)
The answer is 0.128.
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1 solution
Let x = distance between the lamppost's base and Alex. As the angle A is between the camera and the lamppost, the quotient x/20 = tan(A) or x = 20tan(A)
Differentiate with respect to time:
dx/dt = 20sec^2(A)[dA/dt]
dA/dt =(1/20)cos^2(A)[dx/dt]
dx/dt = 4
dA/dt = (1/5)cos^2(A)
When x is 15, by the Pythagorean Theorem, the beam length is 25
Therefore, cos (A) = 20/25 = 4/5
cos^2(A) =(4/5)^2 = 16/25.
(1/5)(16/25) = 16/125 = 0.128 radians per second