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Suppose that 32 feet of fencing is available to enclose a rectangular garden., one side of which will be against the side of a house. What dimensions of the garden will guarantee a maximum area?

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  1. A = x*y
    32 = x+2y

    x = 32-2y
    A = (32-2y)y = 32y-2y^2
    or
    y^2-16y = -A/2
    you did not say if you are in algebra or calculus so I will do it the algebra way by completing the square
    y^2 -16y + (16/2)^2 = -A/2 + 64

    (y-8)^2 = -(1/2)(A-128)
    y = 8
    x = 16
    and the area is 128 ft^2

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  2. area=LW= but L=32-2w
    area=W(32-2W)=32w-2w^2
    but roots of the equation are
    0=2w(16-w), w=0, or w=16

    since this is a parabola, the max/min will occur at 8 (halfway between the roots).
    so w=8, L=16

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