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Calculus

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A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100m^2. what are the dimensions of each pen that minimize the amount of fence that must be used?

  • Calculus - ,

    If each pen has width x and length y (against the barn),
    the area is 4xy, and the fence used is 5x+8y

    So, each pen has area 100.

    p = 5x + 8(100/x) = 5x + 800/x
    dp/dx = 5 - 800/x^2

    dp/dx = 0 when x = √160 = 4√10

    x = 4√10
    y = 25/√10

  • Calculus - ,

    Now, if the barn is used as one wall of the pen, meaning only 3 sides have to be fenced, then

    p = 5x+4y = 5x + 4(100/x) = 5x + 400/x
    p' = 5 - 400/x^2

    p' = 0 at x = 4√5
    y = 25/√5

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