Problema Solution
A farmer has 2000 feet of fencing available to enclose a rectangular area bordering a river. If no fencing is required along the river, find the dimensions of the fence that will maximize area. What is the maximum area?
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let
l = the length of the rectangular area
w = the width of the rectangular area
a farmer has 2000 feet of fencing available to enclose a rectangular area bordering a river
l + 2w = 2000
l = 2000 - 2w
the area is A = l*w
plug l = 2000 - 2w into A = l*w
A = (2000 - 2w)*w
we need to find the maximum of the function A = - 2w^2 + 2000w
A max = 0 - 2000^2/(4*(-2))
A max = 500,000 ft^2
by solving - 2w^2 + 2000w = 500000 we find
w = 500 ft
click here to see the step by step solution of the equation
l = 2000 - 2*500
l = 1000 ft
the dimensions that will maximize the area are length = 1000 ft and width = 500 ft.
the maximum area is 500,000 ft^2.