A rectangle has a perimeter of 80cm. if its width is x, express its length and its area in the terms of X. What is the maximum area of the rectangle?

Perimeter of a rectangle=2 (length) + 2 (width)

length=y

width=x

Then:

80=2x+2y

40=x+y

y=40-x

Then:

width=x

length = (40-x)

Area of a rectangle=length x width.

A (x) = x (40-x)

A (x) = 40x-x²

We have to find the maximums or minimums of this function

1) we calculate the first derivative:

A' (x) = 40-2x

2) we have to find the values of "x" when A' (x) = 0

40-2x=0

-2x=-40

x=-40/-2

x=20

3) We have to calculate the second derivative.

A'' (x) = - 2

Because A'' (x) <0; then we have a maximum at x=20

Therefore:

width: x=20

length: 40-x=40-20=20

This rectangle with maximum area is a square:

Area = (20 cm) (20 cm) = 400 cm²

Answer: The lenght expressed in the terms of x would be: 40-x; and the maximun area would be 400 cm².