Answer the following question.

Question

A patrol car is parked 40 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of 28 revolutions per minute. How fast is the light beam moving (in ft/sec) along the wall when the beam makes angles of 𝜃 = 45°, 𝜃 = 60°, and 𝜃 = 85° with the line perpendicular from the light to the wall? (Round your answers to two decimal places.)

(a)    𝜃 = 45°

(b)    𝜃 = 60°

(c)    𝜃 = 85°

Answer 1

So let's solve this for arbitrary theta, and you can plug in the values.

This is a related rate problem where we want to relate dθ/dt to dx/dt. the procedure usually involves

1) finding an equation that relates θ and x

2) finding a way to get dθ/dt

3) Differentiating the relationship between θ and x with t and relating the rates

4) solving for dx/dt generally and at certain t or θ

1) Looking at the geometry: x/y = tanθ or x = 40tanθ in feet

2) You are given that the police light rotates at 28 revs/minute = 2π*28 rads/60s = (14/25) π rad/s

3) dx/dt = 40(sec²θ)dθ/dt Using chain rule for d/dt (tanθ)

4) The problem gives θ in degrees. I think it's best to convert to radians degrees (π/180) = radians and make sure your calculator is in radians. Plug into the RHS and you have your answer.