Anthony A. answered • 02/18/18

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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