calculus

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The beacon on a lighthouse makes one revolution every 20 seconds. The beacon is 300 feet from
the nearest point, P, on a straight shoreline. Find the rate at which the ray of light moves along
the shore at a point 200 feet from P.

  • calculus -

    Make a sketch, let the distance between the end of the ray of light and point P be x ft
    let the angle formed at that moment be Ø
    then,
    tanØ = x/300
    x = 300tanØ
    dx/dt =300sec^2 Ø dØ/dt

    let the hypotenuse be h
    when x=200
    h^2 = 200^2+300^2
    h = 100√13
    secØ = 100√13/300 = √13/3
    sec^2 Ø = 13/9
    and we are told dØ/dt = 2π/20 rad/sec
    = π/10 rad/sec

    dx/dt = 300(13/9)(π/10) = 130π/3 ft/sec

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