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A Farmer Wants to Maximize Area

Q: A farmer wants to put a fence around a vegetable garden. Only three sides must be fenced, since a rock wall will form the fourth side. If he uses 40m of fencing what is the maximum area possible?

A:  Okay, so we have two widths and one length.

The material used would be:

40 = 2w + L

Note that this is different from the regular perimeter formula, P=2w+2L.

Rearrange it to isolate for L:

L = 40 – 2w

We know that the formula for Area is:

A=L*W, and we are trying to maximize this, so we will work with the Area
formula.

Now substitute the L=40-2w expression into the Area equation:

A = L*w
A = (40-2w)*w

Now complete the square and turn this formula into parabola form:

A= (40-2w)*w
A = 40w -2w^2
A = -2w^2 + 40w
A = -2(w^2 -20 + 100) + 200
A = -2(w-10)^2 + 200

Graph this (by hand, with a graphing calculator, or with Wolfram Alpha) to find that the maximum area is 200, and it occurs at w=10.

So the maximum area is 200 m^2, and the dimensions would be w=10, L=20.

Hope this helps!