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Math Help - elliptical word problem

  1. #1
    NKS
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    elliptical word problem

    A bridge is built in the shape of a semielliptical arch. It has a span of 106 feet. The height of the arch 25 feet from the center is 5 feet. Find the height of the arch at its center.

    Now I'm pretty sure what I can do is 25^2/11236 + 5^2/b^2 = 1
    so.. 625/11236 + 25/b^2 = 1
    then solve for b

    is this correct??
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  2. #2
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    Hello, NKS!

    A bridge is built in the shape of a semielliptical arch.
    It has a span of 106 feet.
    The height of the arch 25 feet from the center is 5 feet.
    Find the height of the arch at its center.
    Code:
                      |
                    * o *
              *       |       o
           *          |       :  *
          *           |b      :   *
                      |       :5
         *            |       :    *
        -*- - - - - - + - - - o - -o - -
       -53            |      25   53

    We have: . \frac{x^2}{53^2} + \frac{y^2}{b^2} \:=\:1 \quad\Rightarrow\quad b^2 \:=\:\frac{53^2y^2}{2809 - x^2}

    Since (25,5) is on the ellipse: . b^2 \:=\:\frac{53^25^2}{2809 - 25^2} \;=\;\frac{53^25^2}{2184}

    . . b \;=\;\frac{53\cdot5}{\sqrt{2184}} \:=\:\frac{265}{2\sqrt{546}} \:=\:\frac{265\sqrt{546}}{1092}


    The height is about 5.67 feet.

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  3. #3
    NKS
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    thanks so much! it seems I did set up the problem correctly. I also came to the same answer
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  4. #4
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    I read that I wasn't suppose to add problems to a thread that is not mine; however, how about if it's similar?

    My problem is the the one on here. Please do excuse me if I'm not suppose to do this, but I would really appreciate your help.

    A bride is to be built in the shape of a semielliptical arch and is to have a span of 100 feet. The height of the arch, at a distace of 40 from the center, is to be 10 feet. Find the height of the arch at its center.

    I have come to the conclusion that b=16.66

    So, if I would like to go further, like search for its eccentricity, I would have to find e=c/a

    Which in this case are a=50, and c=47.14280857

    This gives me that e=.9428561716, which is in fact smaller than one, making it an elllipse. Please check if my answers are correct. I have been working on this problem for weeks now.
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