Question

the figure shows a rectangular pool surrounded by a concrete path that is x feet(ft) wide on all sides. the pool is 21 long and 13 ft wide. the area of the concrete path is 240 ft². what is the value of x? a. 3 b. 6 c. 20 d. 40 note: figure not drawn to scale

Explanation:

Step1: Find outer - rectangle dimensions

The length of the outer - rectangle is $21 + 2x$ (pool length plus $x$ on each side), and the width of the outer - rectangle is $13+2x$ (pool width plus $x$ on each side).

Step2: Calculate area of outer - rectangle and pool

The area of the outer - rectangle $A_{outer}=(21 + 2x)(13 + 2x)$. The area of the pool $A_{pool}=21\times13 = 273$ square - feet.

Step3: Set up equation for area of path

The area of the path is the area of the outer - rectangle minus the area of the pool. So, $(21 + 2x)(13 + 2x)-273 = 240$.
Expand the left - hand side:
\[

$$\begin{align*} (21 + 2x)(13 + 2x)-273&=240\\ 273+42x + 26x+4x^{2}-273&=240\\ 4x^{2}+68x&=240\\ x^{2}+17x - 60&=0 \end{align*}$$

\]

Step4: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 1$, $b = 17$, $c=-60$), we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ or factor the equation.
Factor $x^{2}+17x - 60=(x + 20)(x - 3)=0$.
So, $x=-20$ or $x = 3$. Since $x$ represents a width, it cannot be negative.

Answer:

a. 3