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# 3GTG 5

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Brian has 100 feet of fencing. He will use the fencing to enclose a play area for his puppy. What is the maximum number of square feet he can enclose? Express your answer to the nearest whole number.

Dec 9, 2018

#1
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This is actually a pretty neat problem as we don't seem to be restricted to rectangles.

Do you know which planar shape has the largest area to perimeter ratio?

Think about it and when you figure it out the answer to this is pretty simple.

Dec 9, 2018
#2
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For any perimeter,P,  the largest area that can be enclosed is a square with a side of P/4

So....the max area is

(100/4)^2 =  25^2   =   625 ft^2

Dec 9, 2018
#3
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Suppose we make a circular fence.

$$C = 100 = 2\pi r\\ r=\dfrac{C}{2\pi}\\ A = \pi r^2 = \pi \left(\dfrac{C}{2\pi}\right)^2 = \\ \dfrac{100^2}{4\pi} = \dfrac{2500}{\pi} \approx 796~ft^2$$

Rom  Dec 9, 2018
edited by Rom  Dec 9, 2018
edited by Rom  Dec 9, 2018
#4
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Ah....thanks, Rom....I didn't think about that   !!!

CPhill  Dec 9, 2018
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Wow, You can also make a circle. I had no idea.

dgfgrafgdfge111  Dec 9, 2018
edited by dgfgrafgdfge111  Mar 13, 2019