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Math Help - Trig Word Problem

  1. #1
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    Trig Word Problem

    A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 25 degrees.
    How high (in feet) is the ballon?
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  2. #2
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    Quote Originally Posted by qbkr21 View Post
    A hot-air balloon is floating above a straight road. To calculate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 23 degrees and 25 degrees.
    How high (in feet) is the ballon?
    X, dinstance from previous milepost (measured at ground level)
    Y, dinstance to new milepost (measured at ground level)

    h = X * tan(23)
    h = Y * tan(25)
    X + Y = 1 mile

    A linear equation system.

    Edit: Oh, and according to google, 1 mile = 5 280 feet
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  3. #3
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    Hello, qbkr21!

    A hot-air balloon is floating above a straight road.
    To calculate their height above the ground, the balloonists simultaneously
    measure the angle of depression to two consecutive mileposts
    on the road on the same side of the balloon.
    The angles of depression are found to be 23 and 25.
    How high (in feet) is the ballon?

    The observation to the nearer milepost looks like this:
    Code:
        B * - - - - - - - - - - - - - - -
          : o 25
          :   o
          :     o
          :       o
          :         o
          :           o
          :         25 o 
        --+---------------o---------------*--
                          : - - - 1 - - - :

    The observation to the further milepost looks like this:
    Code:
        B * - - - - - - - - - - - - - - -
          :   * 23
          :       *
          :           *
          :               *
          :                   *
          :                       *
          :                       23 *
        --+---------------o---------------*--
                          : - - - 1 - - - :

    Combining the diagrams we have:
    Code:
        B *
          : o *
          :   o   *
          :     o     *
        y :       o       *
          :         o         *
          :           o           *
          :         25 o         23 *
        --+---------------o---------------*--
          : - - - x - - - : - - - 1 - - - :

    We have: . \tan25^o \,=\,\frac{y}{x}\quad\Rightarrow\quad y \,=\,x\!\cdot\!\tan25^o [1]

    And: . \tan23^o \,= \,\frac{y}{x+1}\quad\Rightarrow\quad y \,=\,(x+1)\tan23^o [2]

    Equate [1] and [2]: . x\tan25^o \:=\:(x + 1)\tan23^o
    . . . . . . . . . . . . . . . x\tan25^o \:=\:x\tan23^o + \tan23^o
    . . . . . . . x\tan25^o - x\tan23^o \:=\:\tan23^o
    . . . . . . . x\left(\tan25^o - \tan23^o\right) \:=\:\tan23^o
    . . . . . . . . . . . . . . . . . . x \:=\:\frac{\tan23^o}{\tan25^o - \tan23^o} \:\approx\:10.147

    Substitute into [1]: . y \:=\:(10.147)\tan25^o \:=\:4.73162.. miles


    Therefore, the height of the balloon is about \boxed{4.7\text{ miles.}}

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  4. #4
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    Re:

    Ok guys thanks, but I already got the 4.7. I was easy I just chose to solve for Y. However the computer system still isn't accepting the answer. It gives me the following hints:

    Hint: 1. Did you convert miles into feet?
    2. Did you convert degrees to radians?

    Taking 4.7 and mulitplying by (5280) wont work, any suggestions.

    What should I do from here?

    Thanks Alot for you help,
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  5. #5
    Senior Member TriKri's Avatar
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    A can se that I and Soroban (and probably you to qbkr21), have caught the problem in two different ways, I thought the baloon where between the to mileposts, while Soroban thought the two mileposts where both in front of the baloon. Maybe that's the problem?
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  6. #6
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    Hello again, qbkr21!

    However the computer system still isn't accepting the answer.
    It gives me the following hints:

    1. Did you convert miles into feet?
    . . Maybe they want more accuracy?
    . . They should have specified the number of decimal places.

    2. Did you convert degrees to radians?
    . . Whatever for? .What difference can it possibly make?

    Try keeping all the decimal places . . .

    x \:=\:10.14692754

    y \:=\:(10.14692754)(\tan25^o) \:=\:4.731590021 miles

    y \:=\:(4.731590021\text{ miles}) \times 5280 \:=\:\boxed{24,982.79531\text{ feet}}

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  7. #7
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    Re:

    BINGO, BINGO, BINGO!!!!! YOU HIT JACKPOT!!!


    Thanks so much for the help,

    All of you are great!!!
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