A patrol car is parked 70 feet from a long warehouse (see figure).

Question

The revolving light on top of the car turns at a rate of 36 revolutions per minute. How fast is the light beam moving (in ft/sec) along the wall when the beam makes angles of 𝜃 = 30°, 𝜃 = 60° and 𝜃 = 85° with the line perpendicular from the light to the wall? (Round your answers to two decimal places.)

(a)    𝜃 = 30°

?ft/sec

(b)    𝜃 = 60°

?ft/sec

(c)    𝜃 = 85°

?ft/sec

Answer 1

Hi Kira,

For this problem we can relate x, y, and Θ using tangent.

tan(Θ) = x/y

Since y is a constant of 70, we can state that:

tan(Θ) = x/70

We're interested in the rate of change of x, or dx/dt. We can implicitly differentiate this equation with respect to time to generate dx/dt.

d/dt (tan(Θ) = x/70)

sec²(Θ) * dΘ/dt = 1/70 * dx/dt

dx/dt = 70 * sec²(Θ) * dΘ/dt

dΘ/dt is the rate of change of the angle. The problem states that the beam revolves 36 times per minute. Since there are 2π radians per revolution, that means:

dΘ/dt = 36 revolutions/ min * 2π radians/ revolution = 72π rad/ minute.

Eventually you'll want ft./sec, so you'll need to convert this to rad/ second.

dΘ/dt = 72π rad/ minute * 1 minute/ 60 sec = 6π/5 rad/ sec

So,

dx/dt = 70 * sec²(Θ) * 6π/5

dx/dt = 84π * sec²(Θ)

Now you just plug in values.

a) 30 deg = π/6 rad

dx/dt = 84π * sec²(π/6) = 84π * (2/√3)² = 112π ft/ sec

b) 60 deg = π/3 rad

dx/dt = 84π * sec²(π/3) = 84π * (2/1)² = 336π ft/ sec

c) 85 deg

Did you mean 75? Because I think 85 needs a calculator. 75 you can at least use cos(30+45). Not sure how to go about determining 85 manually.

Hope this helps!