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Math Help - Conic Section Application Problems

  1. #1
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    Conic Section Application Problems

    Problem -
    "A semielliptic archway has a height of 20 feet and a width of 50 feet. Can a truck 14 feet high and 10 feet wide drive under the archway without going into the other lane."

    I understand that it pertains to an ellipse, but could it essentially be an upside-down hyperbola? Either way, I'm not sure where to actually start. Hopefully somebody can help me, and thank you.
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  2. #2
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    Hello, HawthorneKitty!

    A semielliptic archway has a height of 20 feet and a width of 50 feet. Can a truck
    14 feet high and 10 feet wide drive under the archway without going into the other lane?
    Code:
                    20|
                  * * * * *
              *       |       *  P
            *         * - - - - o(10,14)
           *          |         |*
                      |14       |
          *           |    10   | *
          * - - - - - * - - - - * *
          :    25     :    25     :

    We have an ellipse with: a = 25,\:b = 20

    Its equation is: . \frac{x^2}{25^2} + \frac{y^2}{20^2} \:=\:1

    Question: Is the upper corner of the truck P(10,14) inside the ellipse?


    Substitute the point into the left side of the equation:

    . . \frac{10^2}{25^2} + \frac{14^2}{20^2} \:=\:0.65\:<\:1


    Since the value is less than 1, point P is inside the ellipse.

    Yes, the truck can drive through without going into the other lane.

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