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From a lighthouse (270m above the ocean), an observer notices a ship. The lighthouse is setback from the ocean about 60m. To keep the ships safe from an underwater obstacle at this spot, they need to stay at least 1.4km away from shore. To see the ship, the observer must look down through an angle of depression of 3.24 degrees. How close is the ship to hitting the obstacle?

  • math - ,

    I assume that by "at this spot" you mean 1.4km from the shore at the base of the lighthouse.

    So, draw a diagram.

    Let T be the top of the lighthouse
    Let B be the base of the lighthouse
    Let S be the point on the shore 60 form B
    Let V be the location of the vessel (or ship, but S was taken)
    Let O be the obstacle

    We want to figure OV = BV - BO

    SO = 1.4km = 1400m
    tan 3.24° = TB/BV = 270/BV, so BV = 4769
    SV = BV-BS = 4769-60 = 4709

    The ship is 4709-60 = 4649m from shore, and thus 4649-1400m or 3249m or 3.25km from the obstacle

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