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Math Help - Rectangular Field Along River

  1. #1
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    Rectangular Field Along River

    Jack has 3000 feet of fencing available to enclose a rectangular field. One side of the field lies along a river, so only 3 sides require fencing.

    (a) Express the area A of the rectangle as a function of x, where x is the length of the side parallel to the river.

    (b) Graph A = A(x). For what value of x is the area largest?
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  2. #2
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    Hello, symmetry!

    Did you make a sketch?


    Jack has 3000 feet of fencing available to enclose a rectangular field.
    One side of the field lies along a river, so only 3 sides require fencing.

    (a) Express the area A of the rectangle as a function of x,
    where x is the length of the side parallel to the river.

    (b) Graph A = A(x). .For what value of x is the area largest?
    Code:
        ~ + ~ ~ ~ ~ ~ ~ + ~
          |             |
         y|             |y
          |             |
          * - - - - - - *
                 x
    The area of the field is: . A \:=\:xy [1]

    The total fencing is: x + 2y.
    Since Jack has 3000 feet of fencing: .  x + 2y \:=\:3000\quad\Rightarrow\quad y \:=\:\frac{3000 - x}{2} [2]

    Substitute [2] into [1]: . A \:=\:x\left(\frac{3000 - x}{2}\right)\quad\Rightarrow\quad\boxed{A \:=\:1500x - \frac{1}{2}x^2} (a)


    If we graph A \:=\:1500x - \frac{1}{2}x^2, it looks like this:
    Code:
            |
            |           *
            |      *    :    *
            |   *       :       *
            | *         :         *
            |*          :          *
            |           :
          --*-----------+-----------*--
            |         1500        3000
    The maximum A occurs when x = 1500.

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  3. #3
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    ok

    Soroban,

    This reply is one of the best yet on this site. I love the way you break things down step by step.

    Thanks!
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