Question

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?

Answer 1

Use the conditions expressed in the question to form a quadratic equation and solve to find the lengths of the shortest (`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 `(41 - 2t)/2`

The area is:

`t * (41-2t)/2 = 91`

Multiply both sides by `2` to get:

`182 = 41t - 2t^2`

Subtract the right hand side from the left to get:

`2t^2-41t+182 = 0`

Use the quadratic formula to find:

`t = (41+-sqrt(41^2 - (4xx2xx182)))/(2*2)`

`= (41+-sqrt(1681 - 1456))/4`

`= (41+-sqrt(225))/4`

`= (41+-15)/4`

That is `t = 26/4 = 13/2` or `t = 56/4 = 14`

So the shortest side is length `13/2` inches and the longest is `14` inches