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 = 80° with the line perpendicular from the light to the wall?

Question

A patrol car is parked 65 feet from a long warehouse. The revolving light on top of the car turns at a rate of 26 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 = 80° with the line perpendicular from the light to the wall?

 𝜃 = 45°

𝜃 = 60°

𝜃 = 80° (Round your answer to two decimal places.)

Answer 1

We first generate a diagram to establish the equation relating the variables in question: the angle the beam makes with a perpendicular to the wall and the distance from the beam along the wall to the spot where that perpendicular hits the wall. The distance and the angle are related by right triangle trig, with tanΘ = x / 65, where x is the distance along the wall. We want to calculate dx/dt for the 3 given Θ's which we can do because dΘ/dt is constant, determined by the 26 rpm angular speed of the beam.

We want to convert that angular speed to radians/sec, which we do using dimensional analysis:

dΘ/dt = 26 rev / 1 min · 2π radians / 1 rev · 1 min / 60 sec = 13π/15 rads/sec

65tanΘ = x

65sec²Θ dΘ/dt = dx/dt

Plug in the given values for Θ to calculate those linear speeds.