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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- Mar 9th 2008, 06:52 PMolaitanmaximize 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.

- Mar 9th 2008, 11:04 PMroy_zhang
Let the dimension for the space be by , where denotes length and denotes width.

From given we have , also the area can be expressed as . Our goal is to maximize the area .

Using the given perimeter constraint, we can write . Then let's rewrite the area as:

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

which implies that when , we obtain maximum area . Thus the dimension should be .