# How do you find the dimensions and maximum area of a rectangle whose perimeter is 24 inches?

Feb 7, 2016

A $6 \times 6$ square with area 36 square inches

#### Explanation:

The rectangle has sides h and l, perimeter p and area a:

(a) $p = 2 h + 2 l = 24$

(b) $a = h \cdot l$

(a) $h = 12 - l$

(b) $a = \left(12 - l\right) \cdot l = 12 l - {l}^{2}$

The maximum of $a$ occurs when the first derivative is zero:

$\left(12 l - {l}^{2}\right) ' = 12 - 2 l = 0$
$l = \frac{12}{2} = 6$

$h = 12 - 6 = 6$