# A rectangular lawn is 24 feet wide by 32 feet long. A sidewalk will be built along the inside edges of all four sides. The remaining lawn will have an area of 425 square feet. How wide will the walk be?

Sep 27, 2015

$\text{width" = "3.5 m}$

#### Explanation:

Take the width of the side walk as $x$, so the length of the remaining lawn becomes

$l = 32 - 2 x$

and the width of the lawn becomes

$w = 24 - 2 x$

The area of the lawn is

$A = l \cdot w = \left(32 - 2 x\right) \cdot \left(24 - 2 x\right) = 4 {x}^{2} - 112 x + 768$

This is equal to ${\text{425 ft}}^{2} \to$ given

This means that you have

$4 {x}^{2} - 112 x + 768 = 425$

$4 {x}^{2} - 112 x + 343 = 0$

This is a quadratic equation and you can solve it using the quadratic formula

${x}_{1 , 2} = \frac{- b \pm \sqrt{{b}^{2} - 4 \cdot a \cdot c}}{2 \cdot a} \text{ }$, where

$a$ is the coefficient of ${x}^{2} \to$ $4$ in this case
$b$ is the coefficient of $x \to$ $- 112$ in this case
$c$ is the constant $\to 343$ in this case

Out of the two values which you get for $x$, one will be absurd. Discard it and consider the other.

${x}_{1 , 2} = \frac{- \left(- 112\right) \pm \sqrt{7056}}{2 \cdot 4}$

${x}_{1 , 2} = \frac{112 \pm 84}{8} = \left\{\begin{matrix}\textcolor{red}{\cancel{\textcolor{b l a c k}{{x}_{1} = 24.5}}} \\ {x}_{2} = 3.5\end{matrix}\right.$

Thewidth of the sidewalk will thus be

$x = \text{3.5 m}$