# You are planning a rectangular patio with length that is 7 ft less than three times its width. The area of the patio is 120 ft^2. What are the dimensions of the patio?

Oct 21, 2016

The dimensions of the rectangular patio are width = $7.67$ and the length = $16.01$ ft.

#### Explanation:

Since the length is defined by the width, let $x$ represent the width. This means that the length will be represented by the expression $3 x - 7$. The area of a rectangle is found by multiplying its length by its width. Substitute and solve for $w$.

$A = l w$
$120 = \left(3 x - 7\right) x$
$120 = 3 {x}^{2} - 7 x$
$120 - 120 = 3 {x}^{2} - 7 x - 120$
$0 = 3 {x}^{2} - 7 x - 120$

Now that the equation is simplified and in standard form, use the Quadratic Formula to find the possible solutions for $x$. The Quadratic Formula is $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$. For this situation, $a = 3$, $b = - 7$, and $c = - 120$.

$x = \frac{- \left(- 7\right) \pm \sqrt{{\left(- 7\right)}^{2} - 4 \left(3\right) \left(- 120\right)}}{2 \cdot 3}$
$x = \frac{7 \pm \sqrt{49 + 1440}}{6}$
$x = \frac{7 \pm \sqrt{1489}}{6}$
$x \approx \frac{7 \pm 39}{6}$
$x \approx \frac{7 + 39}{6}$ or $x \approx \frac{7 - 39}{6}$
$x \approx \frac{46}{6}$ or $x \approx - \frac{32}{6}$
$x \approx 7.67$ or $x \approx - 5.33$

Since distances (width, in this case) are not negative, disregard $- 5.33$. The width of the rectangular patio is approximately $7.67$. Use this value to find the length of the patio.

$l = 3 \left(7.67\right) - 7$
$l = 23.01 - 7$
$l = 16.01$

The dimensions of the rectangular patio are width = $7.67$ ft and length = $16.01$ ft.