SOLUTION: A rectangular pen has twelve enclosures. If you have 1,000 feet, what is the maximum area that can be enclosed

Question 1040615: A rectangular pen has twelve enclosures. If you have 1,000 feet, what is the maximum area that can be enclosed
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
If you have 3 enclosures, you need 4 sets of fencing per pen. So for 12 enclosures, you need 13 sets.
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Let the length= L
13L fence is needed.
The width is (1000-13L)/2, because there are two widths.
The area is maximized
L*(1000-13L)/2
500L-(13L/2) is fence used
Take the derivative and set it equal to 0.
500-13L=0
13L=500
L=38.46 feet.
The width is 250 feet.
Twice 250 is 500 feet of fence for both widths.
13L=500 feet.
The area is 500*250=125,000ft^2
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Check using 40 feet of fencing for enclosure. The area should be a little less.
That is 520 feet length and 240 feet width. Area is 124,800 sq ft.
Typically these problems either turn out to be squares or rectangles where the width is half the length.
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