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Math Help - Bearings

  1. #1
    Member purplec16's Avatar
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    Bearings

    From his hotel room window on the fifth floor, Sarano notices some window washers high above on the hotel across the street. Curious as to their height above ground, he quickly estimates the building are 50 ft. apart, the angle of elevation to the workers is about 80 degrees and the angle of depression to the base of the hotel is about 50 degrees.

    How high above ground is the window of Sarano's hotel room?

    How high above ground are the workers?
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  2. #2
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    Hello, purplec16!

    Did you make a sketch?
    This is simple right-triangle Trigonometry.


    From his hotel room window on the fifth floor,
    Sarano notices some window washers high above on the hotel across the street.
    Curious as to their height above ground, he quickly estimates the building are 50 ft. apart,
    the angle of elevation to the workers is about 80 degrees
    and the angle of depression to the base of the hotel is about 50 degrees.

    (a) How high above ground is the window of Sarano's hotel room?

    (b) How high above ground are the workers?
    Code:
                      o W
                    * |
                  *   |
                *     |y
              *       |
            * 80     |
        S * - - - - - * C
          |  * 50    |
         h|     *     |h
          |    50 *  |
        A o - - - - - o B
               50

    Sareno is at S.
    He is h feet above the ground: SA = h

    The hotel is 50 feet away: AB = 50

    Sareno's horizontal line-of-sight is: . SC = 50,\;\;\angle CSB = 50^o

    The workers are at W\!:\;\;\angle W\!SC = 80^o,\;\;y = WC


    In right triangle SAB,\:\angle SAB = 50^o
    We have: . \tan50^o \:=\:\frac{h}{50}\quad\Rightarrow\quad h \;=\;50\tan50^o \;=\;59.58767963 \;\approx\;60

    Therefore, (a) Sareno is 60 feet above the ground.


    In right triangle WCS\!:\;\tan80^o \:=\:\frac{y}{50} \quad\Rightarrow\quad y \;=\;50\tan80^o \;=\;283.564091 \;\approx\;284

    . . W\!B \;=\;h + y \;=\;60 + 284 \;=\;344

    Therefore, (b) the workers are 344 feet above the ground.

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