A school wishes to enclose its rectangular playground using 480 meters of fencing.
Suppose that a side length (in meters) of the playground is , as shown below.

(a) Find a function that gives the area A(x) of the playground (in square meters) in terms of x.

(b) What side length x gives the maximum area that the playground can have?

(c) What is the maximum area that the playground can have?

Part a)

Part b) The side length x that give the maximum area is 120 meters

Part c) The maximum area is 14,400 square meters

Step-by-step explanation:

The picture of the question in the attached figure

Part a) Find a function that gives the area A(x) of the playground (in square meters) in terms of x

we know that

The perimeter of the rectangular playground is given by

we have

substitute

solve for W

Find the area of the rectangular playground

The area is given by

we have

substitute

Convert to function notation

Part b) What side length x gives the maximum area that the playground can have?

we have

This function represent a vertical parabola open downward (the leading coefficient is negative)

The vertex represent a maximum

The x-coordinate of the vertex represent the length that give the maximum area that the playground can have

Convert the quadratic equation into vertex form

Factor -1

Complete the square

The vertex is the point (120,14,400)

therefore

The side length x that give the maximum area is 120 meters

Part c) What is the maximum area that the playground can have?

we know that

The y-coordinate of the vertex represent the maximum area

The vertex is the point (120,14,400) -----> see part b)

therefore

The maximum area is 14,400 square meters

Verify

The playground is a square