# The perimeter of a rectangle is 41 inches and its area is 91 square inches. How do you find the length of its shortest side?

Jul 18, 2015

Use the conditions expressed in the question to form a quadratic equation and solve to find the lengths of the shortest ($\frac{13}{2}$ inches) and longest ($14$ inches) sides.

#### Explanation:

Suppose the length of one side is $t$.

Since the perimeter is $41$, the other side length is $\frac{41 - 2 t}{2}$

The area is:

$t \cdot \frac{41 - 2 t}{2} = 91$

Multiply both sides by $2$ to get:

$182 = 41 t - 2 {t}^{2}$

Subtract the right hand side from the left to get:

$2 {t}^{2} - 41 t + 182 = 0$

Use the quadratic formula to find:

$t = \frac{41 \pm \sqrt{{41}^{2} - \left(4 \times 2 \times 182\right)}}{2 \cdot 2}$

$= \frac{41 \pm \sqrt{1681 - 1456}}{4}$

$= \frac{41 \pm \sqrt{225}}{4}$

$= \frac{41 \pm 15}{4}$

That is $t = \frac{26}{4} = \frac{13}{2}$ or $t = \frac{56}{4} = 14$

So the shortest side is length $\frac{13}{2}$ inches and the longest is $14$ inches