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Calculus

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A lighthouse is located on a small island 4 km away from the nearest
point P on a straight shoreline. Its light makes 3 revolutions
per minute. How fast is the beam of light moving along the shoreline
when it is 1.7 kilometers from
P

?

  • Calculus - ,

    Draw a diagram. If x is the distance from P, and θ is the angle swept out by the beam, measured from the line joining the lighthouse and P,

    x/4 = tan θ

    dx/4 = sec^2 θ dθ

    dx = 4 sec^2 θ * 3 * 2π

    So, what is sec θ? tan θ = 1.7/4 = 0.425
    sec^2θ = 1+tan^2θ = 1.18

    dx = 4*1.18*3*2π = 88.97 mi/min

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