Related Rates of Change

Question

A searchlight rotates at a rate of 4 revolutions per minute. The beam hits a wall located 14 miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per hour) is this dot moving when the angle θ between the beam and the line through the searchlight perpendicular to the wall is π4? Note that dθ/dt=4(2π)=8π.

Answer 1

Let

P be the point on the wall closest to the light source,

h = 15 miles be the distance of the light source from P,

θ(t) be the angle the beam forms with the line perpendicular to the wall at time t,

dθ/dt = 8π/min be the beam's angular velocity,

x(t) be the distance between the beam's dot on the wall and the point P.

The question is about the velocity of beam's dot on the wall, that is, about

dx/dt.

From trigonometry

x = htan(θ)

Differentiate using chain rule

dx/dt = dx/dθ dθ/dt = h/cos²(θ) dθ/dt

You are supposed to calculate dx/dt for a particular value of θ.

There might be a typo in your question, but just for illustration let me assume θ = π/4.

For that θ,

dx/dt = 15/cos²(π/4) 8π = 60π miles/min