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A ship sailing parallel to shore sights a lighthouse at an angle of 12 degrees from its direction of travel. After traveling 5 miles farther, the angle is 22 degrees. At that time, how far is the ship from the lighthouse?

  • Calculus -

    Let the lighthouse be h miles from shore. Let x be the distance of the ship from a point on shore closest to the lighthouse.

    h/(x+5) = tan 12°
    h/x = tan 22°

    so, equating h,

    (x+5)tan 12° = x tan 22°
    .212(x+5) = .404x
    1.06 = .282x
    x = 3.759 mi

    Now, the distance of the ship from the lighthouse, d, can be found by

    3.759/d = cos 22°
    d = 4.05 mi

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