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March 28, 2017

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A rancher plans to set aside a rectangular region of one square kilometer for cattle and wishes to build a wooden fence to enclose the region. Since one side of the region will run along the road, the rancher decides to use a better quality wood for that side which costs three times as much as the wood for the other sides. What dimensions will minimize the cost of the fence?

  • Calculus - ,

    x y = 1

    c = 3 x + x + 2 y = 4 x + 2 y

    c = 4 x + 2/x

    dc/dx = 4 - 2/x^2
    = 0 for minimum

    4 = 2/x^2
    x^2 = 1/2
    x = 1/sqrt 2 = sqrt 2/2 along road
    y = sqrt 2

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