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24 May, 18:51

A rain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges 90 degrees. What depth will provide maximum cross-sectional are and hence allow more water to flow? what depths will allow at least 16 square inches of water to flow?

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  1. 24 May, 22:26
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    2x + y = 12

    y = 12 - 2x

    A = x * y

    A = x * (12 - 2x)

    A = 12x - 2x^2

    Derive

    dA/dx = 12 - 4x

    Set dA/dx = 0

    0 = 12 - 4x

    4x = 12

    x = 3

    A = x * y

    A = 12x - 2x^2

    A = 12 * 3 - 2 * 3^2

    A = 36 - 2 * 9

    A = 36 - 18

    A = 18

    So 18 square inches is the largest cross-section

    x = 3, y = 6

    Those are your dimensions.

    16 = x * y

    16 = 12x - 2x^2

    2x^2 - 12x + 16 = 0

    x^2 - 6x + 8 = 0

    x^2 - 6x + 9 = 1

    (x - 3) ^2 = 1

    x - 3 = - 1, 1

    x = 2, 4

    2 inches or 4 inches will give you a cross-section with an area of 16

    2 < / = x < / = 4 is the domain
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