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17 July, 19:03

Among all rectangles that have a perimeter of 156, find the dimensions of the one whose area is largest. write your answers as fractions reduced to lowest terms.

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  1. 17 July, 21:25
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    P = 2L + 2W

    We have a perimeter of 156, so we have

    2L + 2W = 156

    Let the length = x

    2x + 2W = 156

    The width is

    2W = 156 - 2x

    W = 78 - x

    The area of a rectangle is A = LW

    A = x (78 - x)

    A = 78x - x^2

    This is an inverted parabola, so there is a maximum value.

    78x - x^2 = 0

    x (78 - x) = 0

    x = 0 or x = 78

    The zeros of the parabola are at x = 0 and x = 78.

    Since the parabola is symmetric over its vertical axis, the maximum values occurs at the x-value in the middle of 0 to 78, which is 39.

    At x = 39, the area has a maximum value.

    L = 39 & W = 39

    It's a square with side measuring 39.
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