Question 813199: A farmer has 3000 feet of wire to enclose a rectangular field. He plans to fence the entire area and then subdivide it by running a fence across the middle. Find the dimensions of the fields that would enclose the maximum area. What is the maximum area?
Can you please help me out? Thanks so much in advance:)
Answer by TimothyLamb(4379)   (Show Source):
You can put this solution on YOUR website! 2L + 2w = 3000
L + w = 1500
L = 1500 - w
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A = Lw
A = (1500 - w)w
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Area is now a quadratic function of width (w):
A(w) = -ww + 1500w + 0
A(w) = (-1)w^2 + 1500w + 0
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the above quadratic equation is in standard form, with a=-1, b=1500, and c=0
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to find the maximum area A(w), plug this:
-1 1500 0
into this: https://sooeet.com/math/quadratic-equation-solver.php
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Answer 1:
the maximum point of the above quadratic equation is: ( 750, 562500 )
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so the maximum area that can be enclosed is: 562500 sq.ft. (the y-coordinate of the maximum point)
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now, to find the L and w dimensions of the maximum area, remember this equation from above:
A(w) = (-1)w^2 + 1500w + 0
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we used that eqn to maximize Area, but as a side effect we also got w for that max area, from the maximum point: ( w=750, A=562500 )
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Answer 2:
w for maximum area = 750
L for maximum area = 1500 - w = 1500 - 750 = 750
so the maximum area has dims: w=750, L=750
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