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Richard is building a rectangular playground from 200 feet of fencing. The fencing must entirely enclose the playground. What is the maximum area of this playground?

 Dec 26, 2018
 #1
avatar+3729 
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Hint: The numbers when closer together have the maximum area.

 Dec 26, 2018
 #2
avatar+3576 
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\(\text{I'm going to assume you can't use calculus}\\ p=200 = 2 \ell + 2w\\ w = \dfrac{200-2\ell}{2} \\ area = \ell \cdot w =\\ \ell \left(\dfrac{200-2\ell}{2}\right) = \\ 100\ell - \ell^2 = \\ -(\ell^2 - 100\ell + 2500 - 2500) = \\ -(\ell -50)^2 + 2500 \\ \text{and this has it's maximum value, }2500 \text{ at }\ell=50,~w=50 \\ \)

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 Dec 27, 2018

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