Questions: A rotating light is located 16 feet from a wall. The light completes one rotation every 2 seconds. Find the rate 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. feet per second

Solution Steps

Let $x$ be the distance of the light projected onto the wall from the perpendicular point. Let $\theta$ be the angle between the light beam and the perpendicular to the wall. Since the light is 16 feet from the wall, we have:

$\tan(\theta) = \frac{x}{16}$

So, $x = 16\tan(\theta)$.

The light completes one rotation ($2\pi$ radians) every 2 seconds. Therefore, the rate of change of the angle $\theta$ with respect to time $t$, denoted as $\frac{d\theta}{dt}$, is:

$\frac{d\theta}{dt} = \frac{2\pi}{2} = \pi$ radians/second.

Differentiate the equation $x = 16\tan(\theta)$ with respect to $t$:

$\frac{dx}{dt} = 16\sec^2(\theta) \frac{d\theta}{dt}$

We are given that $\theta = 20^{\circ} = \frac{\pi}{9}$ radians and we found that $\frac{d\theta}{dt} = \pi$ radians/second. Substitute these values:

$\frac{dx}{dt} = 16\sec^2\left(\frac{\pi}{9}\right) \cdot \pi$

Calculating the numerical value:

$\frac{dx}{dt} \approx 16(1.064)^2\pi \approx 16(1.132)\pi \approx 17.808\pi \approx 55.94$ feet/second.

Final Answer: