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11 November, 18:41

A rancher has 800 feet of fencing to put around a rectangular field and then subdivide the field into 3 identical smaller rectangular plots by placing two fences parallel to one of the field's shorter sides. Find the dimensions that maximize the enclosed area. Write your answers as fractions reduced to lowest terms

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  1. 11 November, 21:29
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    Let x be the shorter side, and y be the longer side

    There would be 4 fences along the shorter side, and 2 fences along the longer side

    4x + 2y = 800

    Rewrite in terms of y:

    y = 400 - 2x

    The area of the rectangular field is

    A = x*y

    Replace Y with the equation above:

    A = x (400 - 2x)

    A = - 2x^2 + 400x

    The area is a parabola that opens downward, the maximum area would occur at the parabola vertex.

    At the vertex

    x = - b/2a

    = - 400/[2 (-2) ]

    = 100

    y = 400 - 2x

    y = 400 - 2 (100)

    y = 400-200

    y = 200

    The dimension of the rectangular field that maximize the enclosed area is 100 ft x 200 ft.
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