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AP Calculus AB

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A farmer wants to create a rectangular pen, which will be divided into six separate sections, as shown in the accompanying diagram. If he has 600 feet of fencing to use, what outside dimensions will maximize the area of the pen?

  • AP Calculus AB -

    1. assign x and y values to the length and width of each box within the pens, and use the formula
    600=9x+8y to isolate the y value
    (i got the 8 and 9 by counting the number of length and width values existed in the diagram)
    you should get y=(600-9x)/8

    2. then, since they asked for the maximum area have
    f(x)=x((600-9x)/8)
    and simplify such that f(x)=75x-(9/8)x^2

    3. find the derivative of the function and set it equal to 0
    f(x)=75-(18/8)x or 75-(9/4)x
    x=100/3

    4. then plug the x value into y=(600-9x)/8 to get
    y=75/2

    5. finally count up the number of x and y values found on on the total length and width (3 for x, 2 for y)
    and multiply the numbers by that value
    conveniently (75/2)*2=75
    and (100/3)*3=100

    hope that helps, and i hope that wasn't too late :)

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