feet per second with a rotating light

Question

A rotating light is located 19 feet from a wall. The light completes one rotation every 5 seconds. Find the rate (feet per second) at which the light projected onto the wall is moving along the wall when the light's angle is 20 degrees from perpendicular to the wall.

Answer 1

Let x is distance from projecton of light to the projection of lamp onto the wall. Then x = 19tanθ;

dx/dt = 19sec²θdθ/dt; θ = 20° and dθ/dt = 2π/5 sec^-1.

So, dx/dt = 19sec²20°·2π/5 ft/sec = 27.04 ft/sec