The width of a rectangle is 3 inches less than its length. The area of the rectangle is 340 square inches. What are the length and width of the rectangle?

Apr 19, 2016

Length and width are 20 and 17 inches, respectively.

Explanation:

First of all, let us consider $x$ the length of the rectangle, and $y$ its width. According to the initial statement:

$y = x - 3$

Now, we know that the area of the rectangle is given by:

$A = x \cdot y = x \cdot \left(x - 3\right) = {x}^{2} - 3 x$

and it is equal to:

$A = {x}^{2} - 3 x = 340$

So we get the quadratic equation:

${x}^{2} - 3 x - 340 = 0$

Let us solve it:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

where $a , b , c$ come from $a {x}^{2} + b x + c = 0$. By substituting:

$x = \frac{- \left(- 3\right) \pm \sqrt{{\left(- 3\right)}^{2} - 4 \cdot 1 \cdot \left(- 340\right)}}{2 \cdot 1} =$
$= \frac{3 \pm \sqrt{1369}}{2} = \frac{3 \pm 37}{2}$

We get two solutions:

${x}_{1} = \frac{3 + 37}{2} = 20$
${x}_{2} = \frac{3 - 37}{2} = - 17$

As we are talking about inches, we must take the positive one.

So:

• $\text{Length" = x = 20 " inches}$
• $\text{Width" = y = x-3 = 17 " inches}$