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Thread: Application of Maxima and Minima

  1. #1
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    Exclamation Application of Maxima and Minima

    Kindly help me with this problem.

    Two post, one 8 ft high and the other 12 ft. high, stand 15 ft. apart. They are to be stayed by wires attached to a single stake at ground level, the wires running to the top of the posts. Where should the stake be placed to use the least amount of wire ?
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  2. #2
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    Start by drawing a diagram. If you have the first post be the 8ft post and the second be 12ft, then you can call the position of the stake from the 8ft post $\displaystyle x$.

    You should notice that once you have drawn the diagram with the posts and the wires, you should end up with 2 right angle triangles.

    The first triangle has base $\displaystyle x$ and height $\displaystyle 8$, so the wire will be $\displaystyle \sqrt{8^2 + x^2}$.

    The second triangle has base $\displaystyle 15 - x$ and height $\displaystyle 12$, so the wire will be $\displaystyle \sqrt{12^2 + (15 - x)^2}$.


    So you are minimising $\displaystyle L = \sqrt{8^2 + x^2} + \sqrt{12^2 + (15 - x)^2}$.

    You will need to simplify this, then take the derivative to find the critical points.
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  3. #3
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    Hello, jasonlewiz!

    Two post, one 8 ft high and the other 12 ft. high, stand 15 ft. apart.
    They are to be stayed by wires attached to a single stake at ground level,
    the wires running to the top of the posts.
    Where should the stake be placed to use the least amount of wire ?

    Here is a diagram for Prove It's excellent solution.


    Code:
                                o C
                               *|
                              * |
                             *  |
        A o                 *   |
          | *              *    | 12
          |   *           *     |
        8 |     *        *      |
          |       *     *       |
          |         *  *        |
        B o - - - - - o - - - - o D
          :     x     P   15-x  :
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