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Math Help - Optimization

  1. #1
    Super Member nycmath's Avatar
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    Optimization

    A farmer plans to fence a rectangular pasture near a river. The pasture must have 180,000 m^2 in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river.

    My Work:

    Along the river there is no measurement. Two sides are x and and the third side is (180,000-2x).

    Area of rectangle is length times width.

    A = x(180,000-2x)

    A prime = 180,000-4x.

    I then set A prime to 0 to find my critical point.

    I found x to be 45,000 but this makes no sense. The correct answer is 600 meters by 300 meters. What am I doing wrong?
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  2. #2
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    Re: Optimization

    180,000 is the area of the field, not the total length of the fence.
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  3. #3
    Super Member nycmath's Avatar
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    Re: Optimization

    Perimeter = 2x+y

    Area = xy

    180,000 = xy

    y = (180000)/x

    P = 2x + (180000)/x

    P' = 2 - (180000)/x^2

    Let P' = 0 and solve for x.

    Then x = 300 meters.

    Plug x into A = xy to find y.

    180000 = 300y

    y = (180000)/300

    y = 600

    I did it! :-)
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