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Math Help - Maximizing Area

  1. #1
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    Maximizing Area

    I got this problem with maximizing area.

    This garden is to be fenced with 120 feet of fencing. What dimensions x and y would give the garden maximum area?



    I know that 3x + 2y = 120, and from that, 3x= 120 - 2y.
    Other than that, I'm not sure how to tackle this one.
    Any help much appreciated.
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  2. #2
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    Quote Originally Posted by leviathanwave View Post
    I got this problem with maximizing area.

    This garden is to be fenced with 120 feet of fencing. What dimensions x and y would give the garden maximum area?



    I know that 3x + 2y = 120, and from that, 3x= 120 - 2y.
    Other than that, I'm not sure how to tackle this one.
    Any help much appreciated.
    Hello,

    the area should be maximized. Therefore the main condition is:

    A = x \cdot y

    Now use the length of the fence as you've done above:

    3x = 120-2y~\iff~x=40-\frac23 y

    Plug in this term into the equation of the area to get the characteristic function:

    A(y)=40y - \frac23 y^2

    The graph of this function is a parabola opening down so the maximum value is at it's vertex which you can calculate by -\frac b{2a}~\implies~ -\frac{40}{-\frac43}=30

    Plug in this value into the term calculating x and you'll get x = 20.

    The maximum area is 600 mē
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