Question

Find the dimensions of the rectangle of maximum area whose perimeter is 16 cm ?

Answer 1

The rectangle of maximum area is a square of side length `l=4` cm

Explanation:

Let `x` and `y` be the lengths of the sides of the rectangle measured in cm.

Then the perimeter is:

`2x+2y = 16`

so that:

`y = 8-x`

The area is then:

`S = x*y = x(8-x) = 8x-x^2`

Find the critical points of the function:

`(dS)/dx = 0`

`8-2x = 0`

`x=4`

and as:

`(d^2S)/dx^2 = -2 < 0`

the critical points is a maximum.

Then the maximum area is obtained when `x=4` and `y=8-x=4`, that is when the rectangle is a square.