Question

a rotating light is located 13 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 lights angle is 15 degrees from perpendicular to the wall. feet per second answer is a positive value. give your answer accurate to at least one decimal place.

Explanation:

Step1: Find angular velocity

The light completes one rotation (\(2\pi\) radians) every \(3\) seconds.
The angular velocity \(\frac{d\theta}{dt}=\frac{2\pi}{3}\text{ rad/s}\)

Step2: Establish relationship between \(x\) and \(\theta\)

Let \(x\) be the distance of the light - spot from the point on the wall closest to the light. We know that \(\tan\theta=\frac{x}{13}\), so \(x = 13\tan\theta\)

Step3: Differentiate \(x\) with respect to \(t\)

Using the chain rule, \(\frac{dx}{dt}=13\sec^{2}\theta\frac{d\theta}{dt}\)

Step4: Substitute \(\theta = 15^{\circ}\) and \(\frac{d\theta}{dt}\)

Since \(\theta=15^{\circ}=\frac{\pi}{12}\text{ rad}\) and \(\sec\theta=\frac{1}{\cos\theta}\), \(\cos(15^{\circ})=\cos(\frac{\pi}{12})=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659\), \(\sec^{2}(15^{\circ})=\frac{1}{\cos^{2}(15^{\circ})}\approx1.072\)
\(\frac{d\theta}{dt}=\frac{2\pi}{3}\)
\(\frac{dx}{dt}=13\times\sec^{2}(15^{\circ})\times\frac{2\pi}{3}\)
\(\frac{dx}{dt}=13\times1.072\times\frac{2\pi}{3}\)
\(\frac{dx}{dt}=\frac{27.872\pi}{3}\approx29.2\)

Answer:

\(29.2\)