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.

Question

A rotating light is located 10 feet from a wall. The light completes one rotation every 4 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.

Answer is a positive value. Give your answer accurate to at least one decimal place.

Answer 1

x = 10tanθ, θ = ωt = 2π/4t = π/2t. So, x = 10tan(π/2t).

Now we have to find derivative dx/dt = 10·π/2·sec²θ =5πsec²20° = 17.79 feet/s

Answer 2

Given dθ/dt = 360/4 = 90 deg/sec

Assume the light is rotating clockwise. Let y be the distance down from the perpendicular that the light beam hits the wall. The distance from the light to the wall is given as 10 ft.

A relationship between the light angle and the y position is

tan(θ) = opposite/adjacent = y/10

Taking the derivative of both sides gives:

sec²(θ)dθ/dt = (dy/dt)/10

solving for dy/dt = 10sec²(θ)dθ/dt = 10(1.132)90 = 1019.2 ft/sec