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Math

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Mr. Rogers wants to fence in a rectangular garden next to a straight section of the Scioto
River. He has 330 feet of fencing to do the entire job. He doesn’t need fencing along the
river, and there is a 4.5 foot wide clearance for a gate at one end.
What are the dimensions of the garden (x and y) with the largest area.

Thanks!

  • Math -

    let the width be x
    and the length be y
    so we are given ....
    2x - 4.5 + y = 330
    y = 334.5 - 2x

    area = xy
    = x(334.5-2x)
    = 334.5x - 2x^2

    d(area)/dx = 334.5 - 4x = 0 for a max area
    4x = 334.5
    x = 334.5/4 = 83.625
    y = 334.5 - 2(83.625) = 167.25


    check:
    167.25+83.625+83.625-4.5 = 330
    area = xy = 13986.28125

    let x = 83 , then y = 168.5
    area = 13985.5 , a bit smaller

    let x = 84 , then y = 166.5
    area = xy = 13986 , also a bit smaller than the above
    answer looks good.

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