Question 530159: A farmer has 5000 feet of fencing available to enclose a rectangular field. What is the maximum area?
Answer by oberobic(2304)   (Show Source):
You can put this solution on YOUR website! For any given perimeter, the maximum area is a square.
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5000/4 = 1250
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1250^2 = 1562500 sq. ft.
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If you're not familiar with the maximum area rectangle being a square, a simple application of calculus will demonstrate.
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Recall, perimeter = 2(length + width). With 5000 feet of fencing, that is the maximum perimeter.
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From this starting point, we recognize that 1/2 the perimeter equals the rectangle's length + its width.
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5000/2 = 2500
so
L + W = 2500
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So the sides can be defined as 'L = x' and 'W = 2500-x'.
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y = area = L*W
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Substitute for L and W.
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y = x*(2500-x)
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y = -x^2 + 2500x
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To find the maximum, take the first derivative and solve it for a value of 0.
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dy/dx = -2x + 2500
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2x = 2500
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x = 1250 = L
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2500-x = 1250 = W
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L = W, so we have a square.
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Done.
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