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: Â 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:
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 , and the dimensions would be w=10, L=20.
Hope this helps!
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