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: Considering a given perimeter P, what equation represents the area, A, in terms of the perimeter P, and the side length x for a rectangular playground, adhering to the principle that for any given perimeter, the rectangle with the greatest area is a square?

Part II: Utilizing the equation derived in Part I, how can a simple equation be formulated to determine the least amount of fencing required for a playground with an area of 256 square yards?

Answer 1

The equation that represents the area of a rectangular playground in terms of the perimeter and side length is A = (P/4 - x) * x. To determine the least amount of fencing required for a playground with an area of 256 square yards, we can use this equation and solve for the side length.

Step-by-step explanation:

Part I: The equation that represents the area, A, of a rectangular playground in terms of the perimeter, P, and the side length, x, is A = (P/4 - x) * x. This equation is derived from the fact that for any given perimeter, the rectangle with the greatest area is a square, so the length and width of the rectangle will both be P/4.

Part II: To determine the least amount of fencing required for a playground with an area of 256 square yards, we can use the equation A = (P/4 - x) * x. If we substitute 256 for A, we can solve for x to find the side length. Once we have the side length, we can calculate the perimeter using the formula P = 4x.