To find the area of a regular pentagon-shaped pool, we need to use the area formula for a regular pentagon. Given the side length of 23.62 feet, we proceed as follows:
### Step-by-Step Solution:
1. Identify the side length and number of sides of the pentagon:
- Side length ([tex]\(s\)[/tex]): 23.62 feet
- Number of sides ([tex]\(n\)[/tex]): 5 (since it's a pentagon)
2. Area Formula for Regular Pentagon:
The formula to find the area [tex]\(A\)[/tex] of a regular pentagon with a given side length [tex]\(s\)[/tex] is:
[tex]\[
A = \frac{5}{4} \times s^2 \times \frac{1}{\tan\left(\frac{\pi}{5}\right)}
\][/tex]
3. Plug in the values:
- Side length [tex]\(s = 23.62\)[/tex] feet
[tex]\[
A = \frac{5 \times (23.62)^2}{4 \times \tan\left(\frac{\pi}{5}\right)}
\][/tex]
4. Calculate the tan of [tex]\(\frac{\pi}{5}\)[/tex] and then calculate the entire area:
This requires precise computation to get an exact value (let's not show the intermediary tangent steps as it's detailed).
After performing the calculation:
[tex]\[
A \approx 959.8619118891473 \text{ square feet}
\][/tex]
5. Round to the nearest square foot:
[tex]\[
A_{\text{rounded}} \approx 960 \text{ square feet}
\][/tex]
Thus, the area of the pool that needs to be covered, rounded to the nearest square foot, is:
[tex]\[
\boxed{960 \text{ square feet}}
\][/tex]
