Question

A city park commission received a donation of playground equipment from a parents' organization. The area of the playground needs to be 256 square yards for the children to use it safely. The playground will be rectangular. Part I: For any given perimeter P, the rectangle that encloses the greatest area is a square. Write an equation for the area, A, in terms of the perimeter P, and the side length x. Part II: Use the equation from Part I to result to write a simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards.

Answer 1

Considering the playground to be a square:
Perimeter = x + x + x + x
P = 4x

Area = x²

Px = 4x²
x² = Px/4
Area = Px/4

The fencing will be equal to the perimeter, with area 256 sq yd
256 = Px/4
P = 1024 / x

Answer 2

Answer:

Part 1 : The area in terms of side length =  $x^2$

            Area of square in terms of perimeter P : $(\frac{P}{4})^2$

Part 2: A simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards : $256 = (\frac{P}{4})^2$

Step-by-step explanation:

Part 1 :  For any given perimeter P, the rectangle that encloses the greatest area is a square. . Write an equation for the area, A, in terms of the perimeter P, and the side length x.

Solution :

We are given that the rectangle that encloses the greatest area is a square.

Let the side of the square be x

Let perimeter be P

So, perimeter of square'P' = $4\times side = 4\times x =4x$

So,$P=4x$

Area of square : $(side)^2=x^2$

Thus the area in terms of side length =  $x^2$

Now to find area in terms of perimeter :

Since perimeter $P=4x$

$\Rightarrow x= \frac{P}{4}$

Now, Area of square : $(side)^2$

So,Area of square in terms of perimeter P : $(\frac{P}{4})^2$

Part 2: Use the equation from Part I to result to write a simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards.

Since we have the equation of area in terms of perimeter :

$Area = (\frac{P}{4})^2$

Note: Perimeter tells the amount of fencing

$256 = (\frac{P}{4})^2$

Thus  a simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards : $256 = (\frac{P}{4})^2$