Question

a pool measuring 12 meters by 30 meters is surrounded by a path of uniform width, as shown in the figure. if the area of the pool and the path combined is 1008 square meters, what is the width of the path? the width of the path is \\(\square\\) meters.

Explanation:

Step1: Define the variable

Let the width of the path be \( x \) meters. The length and width of the pool and path combined will be \( (30 + 2x) \) meters and \( (12 + 2x) \) meters respectively (since the path is on both sides of the pool).

Step2: Set up the area equation

The area of a rectangle is length times width. So the area of the pool and path combined is \( (30 + 2x)(12 + 2x) \), and this is given as 1008 square meters. So we have the equation:
\[
(30 + 2x)(12 + 2x)=1008
\]

Step3: Expand the left - hand side

First, expand the product \((30 + 2x)(12 + 2x)\):
\[

$$\begin{align*} (30 + 2x)(12 + 2x)&=30\times12+30\times2x + 2x\times12+2x\times2x\\ &=360+60x + 24x+4x^{2}\\ &=4x^{2}+84x + 360 \end{align*}$$

\]
So our equation becomes \( 4x^{2}+84x + 360 = 1008 \).

Step4: Simplify the equation

Subtract 1008 from both sides to set the equation to zero:
\[
4x^{2}+84x+360 - 1008=0
\]
\[
4x^{2}+84x - 648 = 0
\]
Divide the entire equation by 4 to simplify:
\[
x^{2}+21x - 162=0
\]

Step5: Solve the quadratic equation

We can solve the quadratic equation \( x^{2}+21x - 162 = 0 \) using the quadratic formula \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \), where \( a = 1 \), \( b = 21 \), and \( c=- 162 \).
First, calculate the discriminant \( \Delta=b^{2}-4ac=(21)^{2}-4\times1\times(-162)=441 + 648 = 1089 \)
Then, \( x=\frac{-21\pm\sqrt{1089}}{2\times1}=\frac{-21\pm33}{2} \)
We have two solutions:
\( x_{1}=\frac{-21 + 33}{2}=\frac{12}{2}=6 \)
\( x_{2}=\frac{-21-33}{2}=\frac{-54}{2}=-27 \)
Since the width cannot be negative, we discard \( x=-27 \).

Answer:

The width of the path is \( \boldsymbol{6} \) meters.