Acity 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.

Part 1 : The area in terms of side length =  

            Area of square in terms of perimeter P :

Part 2: A simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards :

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' =

So,

Area of square :

Thus the area in terms of side length =  

Now to find area in terms of perimeter :

Since perimeter

Now, Area of square :

So,Area of square in terms of perimeter P :

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 :

Note: Perimeter tells the amount of fencing

Thus  a simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards :

step-by-step explanation:

step 1: solve one of the equations for one of the variables. let's solve the first equation for y:

step 2: substitute that equation into the other equation, and solve for x.

step 3: substitute x = 4 x = 4 x=4 into one of the original equations, and solve for y.

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step-by-step explanation: