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September 19, 2014

September 19, 2014

Posted by **Kiana** on Tuesday, October 16, 2012 at 11:46am.

- Calculus -
**Steve**, Tuesday, October 16, 2012 at 3:26pmcost is proportional to perimeter, so we want to minimize the perimeter enclosing 108 yd^2. If x is the length of the fence on the property line, 108/x is the other dimension.

p(x) = 2(x + 108/x)

dp/dx = 2(1 - 108/x^2)

dp/dx=0 when x^2 = 108

x = 6√3 = 10.39

the pool is thus

6√3 by 6√3

as we know, a square has maximum area for a given perimeter; conversely a square has minimum perimeter for a given area.

Now, if we want to minimize the cost to the pool owner, that would involve subtracting half the cost of the side on the property line:

p(x) = 2(x + 108/x) - x/2

dp/dx = 2(1 - 108/x^2) - 1/2

dp/dx=0 when x = 12

now the fence would be 12 by 9

The longer border length makes the neighbor pay more.

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