Can anyone help me solve this related rates problem? I have been stuck on it for hours and don't really know how to start it.

Question

A searchlight rotates at a rate of 2 revolutions per minute. The beam hits a wall located 9 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 π/3? Note that dθ/dt=2(2π)=4π. Take note of the units.

Answer 1

Draw a diagram:

We are trying to find dx/dt so that means we need to find a relationship that involves "x" so we can take the derivative with respect to time to get dx/dt.

Notice that in the triangle created, we can use the tangent trig ratio to say:

tan(θ) = x/9 so that means:

x = 9tan(θ)

Taking the derivative, with respect to time, we get (applying the chain rule):

dx/dt = 9sec²(θ)•dθ/dt or

dx/dt = 9[1/cos²(θ)]•dθ/dt or

dx/dt = [9/cos²(θ)]•dθ/dt

We are told that θ = π/3 and cos(π/3) = 1/2 so cos²(π/3) = (1/2)² = 1/4

We are also told that dθ/dt = 4π (the units of this is radians/minute because 1 revolution is 2π radians)

So: dx/dt = [9/(1/4)]•4π = 36•4π = 144π and the units would be miles/minute (9 is in miles and the unit of time is minutes)

(144π miles/min) x (60 min/hr) = 8640π miles/hr

Answer 2

For a given angle θ(t), the position x of the dot on the wall is given by x(t) = D * tanθ(t), where D indicates the distance searchlight - wall [since x(t)/D = tanθ(t), from trigonometry].

The dot speed can be obtained by differentiating wrt time

v(t) = dx(t)/dt = D * (1/cos²θ) * dθ/dt. From the problem, we know dθ/dt. If θ = π/3, then cosπ/3 = 1/2. So,

v(π/3) = D * (1/2^2) * dθ/dt.

dθ/dt = π/60 (in terms of revolutions per hour)