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A farmer wants to make three identical rectangular enclosures along a straight river, as in the diagram shown below. If he has 1680 yards of fence, and if the sides along the river need no fence, what should be the dimensions of each enclosure if the total area is to be maximized?

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  1. If each pen is x by y, with the x side parallel to the river, then

    6x+4y = 1680

    a = 3xy = 3*x(1680-6x)/4 = (2520x-9x^2)/2

    This is just a parabola with vertex (hence maximum area) at x = 140

    So, each pen is 140 by 210

    As usual, the fencing is divided equally between lengths and widths.

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