Calculus

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?

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  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.

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