Question 872829: A field is bounded on one side by a river. A farmer wants to enclose the other three sides of the field with a fence in order to create a rectangular plot of land for his cows. If the farmer has 400m of fence to work with, determine the maximum possible area of the field and the field's dimensions.
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! A field is bounded on one side by a river. A farmer wants to enclose the other three sides of the field with a fence in order to create a rectangular plot of land for his cows. If the farmer has 400m of fence to work with, determine the maximum possible area of the field and the field's dimensions.
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Let x = width
and y = length
2x+y = 400 (eq 1)
xy= area (eq 2)
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Solving eq 1 for y:
y = 400-2x
substitute into eq 2
x(400-2x) = area
-2x^2+400x = area
since the above is a parabola that opens downwards, the vertex is the max.
x-value of the max is:
x = -b/(2a)
x = -400/(2(-2))
x = -400/(-4)
x = 100 feet (width)
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length is:
2x+y=400
2(100)+y=400
200+y=400
y = 200 feet
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Max area is:100*200 = 20000 square feet
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