Related rates

Question

A lighthouse is fixed 170 feet from a straight shoreline. A spotlight revolves at a rate of 17 revolutions per minute, (34 rad/min ), shining a spot along the shoreline as it spins. At what rate is the spot moving when it is along the shoreline 15 feet from the shoreline point closest to the lighthouse?

Answer 1

The rate of revolutions is equivalent to the change in theta over time aka dθ/dt

Imagine a right triangle that has one side being the closest distance (170) and the other being the distance away from that point (x)

The initial angle of the lighthouse θ is tan^-1(15/170) = 5º or π/36

we are looking for the change in shoreline position (x) over the change in time, giving us speed (dx/dt)

now, as theta changes (34 rad/min) so does the ratio, such that tan θ = x/170

we use implicit differentiation with respect to dt to get 

sec^2 (θ)dθ/dt = 1/170 dx/dt

 

solving for dx/dt = 170sec^2(θ)dθ/dt

we have theta and the change in theta

so we plug this in 170*sec^2(π/36)*34 rad/min

this comes out to roughly 5824 ft/min or 97 ft/sec