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Math Help - Enclosing a Rectangualr Field

  1. #1
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    Enclosing a Rectangualr Field

    Beth has 3000 feet of fencing available to enclose a rectangular field.

    a) Express the area A of the rectangle as a function of x, where x is the length of the rectangle.

    b) For what value of x is the area largest?

    c) What is the maximum area?

    I have the answer but do not know how to set up. Can someone help.

    Answers
    a) A(x)=-x^2 + 1500x

    b) A is largest when X = 750ft.

    c) 565,500 ft sq

    Robert
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  2. #2
    MHF Contributor
    Joined
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    One way of doing this is by Analytic Geometry, by the properties of the parabola.
    Another way, an easier way, is by Calculus. I will use this way here.

    Rectangle.
    Length = x ft.
    Perimeter = 3000 ft.
    So,
    Width = (3000 -2x)/2 = 1500 -x ft.

    a) Area, A = x(1500 -x) = 1500x -x^2

    b) A is largest or smallest when dA/dx = 0
    dA/dx = 1500 -2x
    Set that to zero,
    0 = 1500 -2x
    2x = 1500
    x = 1500/2 = 750 ft.
    That should be the x for maximum A. (because minimum A is zero, and here x=0 or x=1500 ft.)

    c) A = 1500x -x^2
    max A = 1500(750) -(750)^2 = 562,500 sq.ft.
    (Not 565,500 sq.ft.)
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