# 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?

##### 1 Answer
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