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Calculus 2

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A farmer wishes to build a fence for 6 adjacent rectangular pens. If there is 600 feet of fencing available, what are the dimensions of each pen that maximizes total pen area?

I keep getting x=150 but I have been told that is not enough fencing. Can anyone help?

  • Calculus 2 -

    You must give a description of the pens.

    Is there a large rectangle with equal partitions parallel to the widths ? (the usual case)

    Since you don't say what the x stood for, I have no way of telling what the 150 represents, since "dimension" implies length and width.

  • Calculus 2 -

    That was the only thing the question said :/. It never gave a description of the pens. There is a picture with 6 boxes connected to each other 3 boxes on top and 3 on bottom:

    box box box
    box box box

    other than that, that was all the info I was given:(

  • Calculus 2 -

    In that case, if each pen has width x and height y in the drawing, then

    total area is 6xy
    Also, 3x+3x+3x+2y+2y+2y+2y = 600, so 9x+8y=600

    a = 6xy = 6x(600-9x)/8
    = 9/4 x(200-3x)
    da/dx = 9/2 (100-3x)
    so, da/dx = 0 when x = 100/3

    so, each small pen is 100/3 by 75/2

    max area = 7500

  • Calculus 2 -

    Ok, then it isn't that bad
    Make a sketch,
    label the length of each small pen as x and its width y
    counting up all the x's and y's, I get
    9x + 8y = 600
    y = (600 - 9x)/8
    where 600-9x > 0
    9x < 600
    x < 66.67

    area = 3x(2y)= 6xy
    = 6x(600-9x)/8
    = 3600x - (27/4)x^2

    This is a parabola which opens downwards, so it has a maximum
    the x of the vertex is -b/(2a) = -3600/(-27/2) =266.67
    which is beyond our restriction of x

    Thus this question has no solution

    by Calculus:
    d(area)/dx = 3600 - 27x/2
    = 0 for a max area
    27x/2 = 3600
    27x = 7200
    x = 266.66..
    y = (600 - 9(266.67)/8 which is a negative
    no solution

    here is a picture of why
    http://www.wolframalpha.com/input/?i=plot+y+%3D+3600x+-+%2827%2F4%29x%5E2

  • Just ignore my last post -Calculus 2 -

    What stupid error that was.

    my mistake is in
    = 6x(600-9x)/8
    = 3600x - (27/4)x^2
    what garbage that is !!!

    Go with STeve

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