Chapter 7.2, Problem 6ST

To determine

To find: The width of the sidewalk.

Answer to Problem 6ST

The width of the sidewalk is 2.387 ft.

Explanation of Solution

Given:

The length of the flower bed = 11 ft.

The width of the flower bed = 13 ft.

The area of the sidewalk = 180 ft².

Calculation:

Consider the constant width of the sidewalk is x .

The total area of the flower bed and the sidewalk is

  $\begin{array}{l}\left(l+x\right)\left(b+x\right)=11\times 13+180\\ \left(11+x\right)\left(13+x\right)=323\\ x^2+24x+80=143\\ x^2+24x-63=0\end{array}$

The solution of the given equation is obtained by using the quadratic formula.

For a quadratic equation in the form of $a{x}^2+bx+c=0$ , the solution of this equation is given by

  $x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Comparing the given equation with the standard quadratic equation, the coefficients are:

  $\begin{array}{l}a=1\\ b=24\\ c=-63\end{array}$

Substituting these in the formula for the roots.

First take the positive sign.

  $\begin{array}{c}{x}_1=\frac{-\left(24\right)\pm \sqrt{{\left(24\right)}^2-4\times 1\times \left(-63\right)}}{2\times 1}\\ =\frac{-24\pm \sqrt{576+252}}{2}\\ =\frac{-24\pm 28.77}{2}\\ =2.387,-26.387\end{array}$

The width cannot be negative.

The width of the sidewalk is 2.387 ft.

Therefore, the width of the sidewalk is 2.387 ft.