Question
A pool measuring 14 meters by 22 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?
14
22
14 + 2x
22 + 2x
.
.
.
Question content area right
Part 1
The width of the path is
enter your response here meters.
14
22
14 + 2x
22 + 2x
.
.
.
Question content area right
Part 1
The width of the path is
enter your response here meters.
Answers
GPT-4o mini
To find the width of the path surrounding the pool, let's denote the width of the path as \( x \) meters.
The dimensions of the pool are:
- Length = 22 meters
- Width = 14 meters
With the path of width \( x \) surrounding the pool, the overall dimensions of the pool plus the path will be:
- Overall Length = \( 22 + 2x \)
- Overall Width = \( 14 + 2x \)
The area of the pool plus the path combined is given to be 1008 square meters. Therefore, we can set up the following equation for the area:
\[
(22 + 2x)(14 + 2x) = 1008
\]
Now, let's expand this equation:
\[
22 \cdot 14 + 22 \cdot 2x + 14 \cdot 2x + 4x^2 = 1008
\]
Calculating \( 22 \cdot 14 \):
\[
308 + 44x + 28x + 4x^2 = 1008
\]
Combining like terms:
\[
4x^2 + 72x + 308 = 1008
\]
Now, let's subtract 1008 from both sides:
\[
4x^2 + 72x + 308 - 1008 = 0
\]
This simplifies to:
\[
4x^2 + 72x - 700 = 0
\]
Now, we can divide the entire equation by 4 to simplify it:
\[
x^2 + 18x - 175 = 0
\]
Next, we can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 18 \), and \( c = -175 \):
Calculating the discriminant:
\[
b^2 - 4ac = 18^2 - 4 \cdot 1 \cdot (-175) = 324 + 700 = 1024
\]
Now, substituting back into the quadratic formula:
\[
x = \frac{-18 \pm \sqrt{1024}}{2 \cdot 1} = \frac{-18 \pm 32}{2}
\]
This gives us two potential values for \( x \):
1. \( x = \frac{14}{2} = 7 \)
2. \( x = \frac{-50}{2} = -25 \) (not a valid solution since width cannot be negative)
Thus, the width of the path is:
\[
\boxed{7} \text{ meters}
\]
The dimensions of the pool are:
- Length = 22 meters
- Width = 14 meters
With the path of width \( x \) surrounding the pool, the overall dimensions of the pool plus the path will be:
- Overall Length = \( 22 + 2x \)
- Overall Width = \( 14 + 2x \)
The area of the pool plus the path combined is given to be 1008 square meters. Therefore, we can set up the following equation for the area:
\[
(22 + 2x)(14 + 2x) = 1008
\]
Now, let's expand this equation:
\[
22 \cdot 14 + 22 \cdot 2x + 14 \cdot 2x + 4x^2 = 1008
\]
Calculating \( 22 \cdot 14 \):
\[
308 + 44x + 28x + 4x^2 = 1008
\]
Combining like terms:
\[
4x^2 + 72x + 308 = 1008
\]
Now, let's subtract 1008 from both sides:
\[
4x^2 + 72x + 308 - 1008 = 0
\]
This simplifies to:
\[
4x^2 + 72x - 700 = 0
\]
Now, we can divide the entire equation by 4 to simplify it:
\[
x^2 + 18x - 175 = 0
\]
Next, we can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 18 \), and \( c = -175 \):
Calculating the discriminant:
\[
b^2 - 4ac = 18^2 - 4 \cdot 1 \cdot (-175) = 324 + 700 = 1024
\]
Now, substituting back into the quadratic formula:
\[
x = \frac{-18 \pm \sqrt{1024}}{2 \cdot 1} = \frac{-18 \pm 32}{2}
\]
This gives us two potential values for \( x \):
1. \( x = \frac{14}{2} = 7 \)
2. \( x = \frac{-50}{2} = -25 \) (not a valid solution since width cannot be negative)
Thus, the width of the path is:
\[
\boxed{7} \text{ meters}
\]