# You have 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area?

Apr 10, 2018

A square of $125 f e e t$ sides
Area = ${125}^{2} = 15625 f e e {t}^{2}$

#### Explanation:

Upon investigation you will discover that the greatest area for any particular circumference is that of a square.

So the length of one side is $\frac{500}{4} = 125 f e e t$
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Lets the lengths of the sides be $b \mathmr{and} c \to 2 b + 2 c = 500$

Thus $b = \frac{500 - 2 c}{2} = 250 - c$

Area$\to a = b \times c \textcolor{w h i t e}{\text{dddd")-> color(white)("dddd}} \left(250 - c\right) \times c$

color(white)("dddddddddd.")" Area "color(white)("d")->color(white)("dddd")a=250c-c^2" "....Eqn(1)

As the ${c}^{2}$ term is negative then the graph is of form $\cap$ thus it has a maximum and this is the vertex.

However, we need to write $E q n \left(1\right)$ in the form of:

$y = 0 = + {c}^{2} - 250 c + a$

using part of the proces of completing the square we have:

${c}_{\text{vertex}} = \left(- \frac{1}{2}\right) \times \left(- 250\right) = + 125$

Substitute this back into $E q n \left(1\right)$

$a = 250 \left(125\right) - \left({125}^{2}\right) = 15625 f e e {t}^{2}$