A rotating light is located 13 feet from a wall. The light completes one rotation every 5 seconds. Find the rate at which the light projected onto the wall is moving along the wall when the light's angle is 10 degrees from perpendicular to the wall.
feet per second
Answer is a positive value. Give your answer accurate to at least one decimal place.
Answer
Final Answer: The rate at which the light projected onto the wall is moving along the wall when the light's angle is 10 degrees from perpendicular to the wall is approximately \(\boxed{16.1}\) feet per second.
Step 1 ：We can model the light's position on the wall using a right triangle. The light is the hypotenuse, the distance from the light to the wall is one leg, and the distance from the light's position on the wall to the point on the wall directly opposite the light is the other leg.
Step 2 ：The light's position on the wall changes as the light rotates, so the length of the leg along the wall is a function of the angle the light makes with the wall.
Step 3 ：Using trigonometry, we find that the cosine of the angle is the adjacent leg (the distance from the light to the wall) divided by the hypotenuse (the light's distance from the point on the wall directly opposite the light). So, the length of the leg along the wall is the light's distance from the point on the wall directly opposite the light times the cosine of the angle.
Step 4 ：The rate at which the light's position on the wall is changing is the derivative of this function with respect to time. We can find this derivative using the chain rule.
Step 5 ：The light completes one rotation every 5 seconds, so the rate at which the angle is changing is \(\frac{2\pi}{5}\) radians per second.
Step 6 ：We can plug in the given values to find the rate at which the light's position on the wall is changing when the light's angle is 10 degrees from perpendicular to the wall.
Step 7 ：The negative sign indicates that the light's position on the wall is moving in the negative direction, which makes sense because the light is rotating. However, the question asks for the rate at which the light's position on the wall is moving, not the direction, so we can ignore the negative sign.
Step 8 ：Final Answer: The rate at which the light projected onto the wall is moving along the wall when the light's angle is 10 degrees from perpendicular to the wall is approximately \(\boxed{16.1}\) feet per second.
