A light in a lighthouse 5 kilometers offshore from a straight shoreline is rotating at 4 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point 5 kilometers from the point opposite the​ lighthouse?

Answer 1

Make a sketch.
Mine has a straight shore line and a lighhouse L and a point A on the shore so that LA is perpendicular to the shore.
I let P be a point on the shore so that AP = x
Let Ø be the angle between LA and LP

given: dØ/dt = 4(2π) radians/min
= 8π rads/min

tanØ = x/5
x = 5tanØ
dx/dt = 5sec^2 Ø dØ/dt
when Ø = 0
dx/dt = 5 (sec^2 0)(8π) km/min
= 5(1)(8π) km/min
=40π km/min

change it to whatever units are needed.