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Math Help - angle depression

  1. #1
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    angle depression

    A light house keeper 120 feet above the water sees a boat sailing in a straight line directly toward her. As she watches, the angle of depression to the boat changes from 28 degrees to 43 degrees. How far has the boat traveled during this time?
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  2. #2
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    Hello, victorfk06!

    A lighthouse keeper 120 feet above the water sees a boat
    sailing in a straight line directly toward her.
    As she watches, the angle of depression to the boat changes from 28 to 43.
    How far has the boat traveled during this time?
    Code:
        A * - - - - - - - - - - - 
          |   *  28
          | 62  * 
          |         *
      120 |             *
          |               *
          |                  *
          |                     *
          |                        *
          * - - - -*- - - - - - - - - *
          B   y    C        x         D
    In right triangle ABD\!:\;\;\tan62^o \:=\:\frac{x+y}{120}\quad\Rightarrow\quad x \;=\;120\tan62^o - y .[1]


    Code:
        A * - - - - - -
          |* 43
          | *
          |  *
      120 |47*
          |    *
          |     *
          |      *
          |       *
          * - - - -*
          B   y    C
    In right triangle ABC\!:\;\;\tan47^o \:=\:\frac{y}{120} \quad\Rightarrow\quad y \:=\:120\tan47^o .[2]


    Substitute [2] into [1]: . x \;=\;120\tan62^o - 120\tan47^o \;=\;97.0029306

    . . Therefore, the boat traveled about 97 feet.

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  3. #3
    Eater of Worlds
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    There are various approaches, but the law of tangents works OK.

    120(tan(62)-tan(47))
    Last edited by galactus; November 24th 2008 at 05:38 AM.
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