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5 May, 22:10

A rancher wants to fence in an area of 2500000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. what is the shortest length of fence that the rancher can use?

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  1. 6 May, 00:31
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    Lets say that,

    x = width of rectangle

    y = length of rectangle

    The area of the rectangle is given as:

    A = x * y = 2500000 ft^2

    Rewriting in terms of y:

    y = 2500000 / x

    The length of fencing needed is equal to the perimeter of rectangle plus the middle fence:

    L = 2 (x + y) + x = 3x + 2y

    Substituting the value of y:

    L = 3x + 2 (2500000/x) = 3x + 5*10^6*x^ (-1)

    The minima can be obtained by taking the 1st derivative of the equation then equating dL/dx = 0:

    dL/dx = 3 - 5*10^6*x^ (-2)

    3 - 5*10^6*x^ (-2) = 0

    x^-2 = 6*10^-7

    x = 1291

    Calculating for y:

    y = 2500000 / x = 2500000 / 1291

    y = 1936.48

    Therefore the shortest length of fence needed is:

    L = 3x + 2y = 3 (1291) + 2 (1936.48)

    L = 7746 ft
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