Question

A rectangular playground is enclosed on three sides by a fence and one side of a house. The length of the fence is 20 yards. Given that the width of the playground is x yards, find:

an expression for the lenth og the playground in temrs of x
the area of the playground in terms of x
the actual length and width of the playground that will give the maximum area

Answer 1

length of playground + 2x = 20
so length of playground = 20 - 2x

area = A = (20 - 2x)x = 20x - 2x^2

dA/dx = 20 - 4x

dA/dx = 0 when x = 5

thus width = 5, length = 10 maximizes the area (maximum area = 50)

Answer 2

or you could find maximum area using the turning point:

A = 20x - 2x^2

= -2x^2 + 20x

= -2(x^2 - 10x)

= -2((x - 5)^2 - 25)

= -2(x - 5)^2 + 50