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Find the rate at which the light projected onto the wall is moving along the wall. Related Rates?

A rotating light is located 10 feet from a wall. The light completes one rotation every 3 seconds. Find the rate at which the light projected onto the wall is moving along the wall when the light's angle is 25 degrees from perpendicular to the wall.

Form a triangle with one side of length 10, a right angle at the end of that side and a 25 degree angle at the other end of this side. The side opposite the 25 degree angle represents the distance x the projected light has moved along the wall.

The relationship between x and the angle theta is: tan theta = x/10. But the light rotates 2pi radians in 3 seconds so theta = 2pi/3*t. Then the equation relating x and t is: 10 tan(2pi/3 *t) = x.

Take the derivative with respect to t: dx/dt = 10*2pi/3*(sec(2pi/3)*t)^2.

Evaluate when theta = 25 degrees = 25pi/180 radians by solving the following equation for t: 25pi/180 = 2pi/3*t. t = 25pi/180*3/2pi = 75/360 = 5/24.

So now find dx/dt when t = 5/24.
At this time the light is moving along the wall at 20pi/3*(sec(2pi/3*5/24))^2.

This is about 25.5 ft/sec.
It was not necessary to find the time above. You could just calculate the speed as 20pi/3 * (sec 25 degrees)^2.

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