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Math Help - Max/min Help!

  1. #1
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    Jul 2007
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    Max/min Help!

    This is yr 12 maths b.A bushwalker can walk at 5km/h through clear land and 3km/h through bushland. If she has to het from point a to point b following a route indicated in the image, find the value of x so that the route is covered in a minimum time. (Note: time = distance/speed)

    Image: http://img294.imageshack.us/my.php?i...ntitledtp0.jpg

    Please help, i need help getting to the answer. Please post all working. thnks
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, mmulv2!

    The set-up is the hard part . . . I'll leave the rest to you.


    A bushwalker can walk at 5 km/h through clear land and 3 km/h through bushland.
    If she has to get from point A to point B following a route indicated in the image,
    find the value of x so that the route is covered in a minimum time.
    Code:
                            * B
                            |
                            | 3-x
                            |
                            * C
                         *  |
             ____     *     |
            √x+4  *        | x
                *           |
             *              |
        A * - - - - - - - - * D
                    2

    She will walk from A to C through bushland.
    . . From right triangle CDA, we have: . AC \:=\:\sqrt{x^2+4} km.
    At 3 km/h, it will take her: . T_1\;=\;\frac{\sqrt{x^2+4}}{3} hours.

    Then she will walk from C to B through clear land.
    . . At 5 km/h, this will take her: . T_2 \;=\;\frac{3-x}{5} hours.

    Hence, her total time is: . T \;=\;\frac{1}{3}(x^2+4)^{\frac{1}{2}} + \frac{3}{5} - \frac{1}{5}x

    . . and that is the function you must minimize.

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