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Math Help - maximize problem

  1. #1
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    Thumbs up maximize problem

    You have 40m of fencing with which to enclose a rectangular space for a garden.find the largest area that can be enclosed with this much fencing and the dimensions of the corresponding garden.
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  2. #2
    Junior Member roy_zhang's Avatar
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    Let the dimension for the space be l by w, where l denotes length and w denotes width.

    From given we have 2l+2w=40, also the area can be expressed as A=l\times w. Our goal is to maximize the area A.

    Using the given perimeter constraint, we can write l=20-w. Then let's rewrite the area as: A=(20-w)w=-w^2+20w

    Now the area is expressed as a function of w and it is a quadratic function whose graph is a parabola open down. Hence it will have a maximum. More specific, we have

    A=-w^2+20w=-(w-10)^2+100

    which implies that when w=10, we obtain maximum area A=100. Thus the dimension should be l=w=10.
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