Question 813196: A farmer has 120 meters of fencing and wants to enclose a rectangular plot of land that requires fencing on only three sides , since it is bounded by a river on one side. Find the length and width of the plot that will maximize the 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! L + 2w = 120
L = 120 - 2w
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A = Lw
A = (120 - 2w)w
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Area is now a quadratic function of width (w):
A(w) = -2ww + 120w + 0
A(w) = (-2)w^2 + 120w + 0
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the above quadratic equation is in standard form, with a=-2, b=120, and c=0
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to find the maximum area A(w), plug this:
-2 120 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: ( 30, 1800 )
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so the maximum area that can be enclosed is: 1800 sq.m (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) = (-2)w^2 + 120w + 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=30, A=1800 )
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Answer 2:
w for maximum area = 30
L for maximum area = 120 - 2w = 120 - 60 = 60
so the maximum area has dims: w=30, L=60
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