A movie crew is working on a scene that involves filming a car moving at a high

Question

A movie crew is working on a scene that involves filming a car moving at a high speed. For one perspective, a camera is positioned and fixed at a spot 50 feet from the car's path (see picture below). Construct a function s(x) that determines the angular velocity (in radians per second) at which the camera should turn to keep the car in frame when the car is x feet from the point O. Assume the car is moving at 90 miles per hour in the positive direction. Be careful with the units. Your function must be in terms of x, not Theta.

Explanation / Answer

tan = x/50
differentiating with respect to t,
sec^2 * d/dt = 1/50 * dx/dt
sec = sqrt(x^2+50^2)/50
we have,
sec^2 * d/dt = 1/50 * dx/dt
(x^2+50^2)/50 * d/dt = 1/50* 90
d/dt = 90 / (x^2+50^2)
so,
s(x) = 90 / (x^2+50^2)

when x = 50
s(50) = 90 / (50^2+50^2

= 1.27

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