To determine the area of a regular pentagon-shaped pool, follow these steps:
1. Identify the Given Information:
- Side length of the pentagon: [tex]\(23.62\)[/tex] feet.
- Radius (distance from the center to a vertex) of the pentagon: [tex]\(20.10\)[/tex] feet.
2. Recall the Formula for the Area of a Regular Pentagon:
The area [tex]\(A\)[/tex] of a regular pentagon with side length [tex]\(s\)[/tex] can be calculated using the formula:
[tex]\[
A = \left( \frac{5}{4} \right) \cdot s^2 \cdot \left( \frac{1}{\tan(\pi/5)} \right)
\][/tex]
This formula uses the tangent of [tex]\(\pi/5\)[/tex], which accounts for the interior angles of the pentagon.
3. Plug in the Given Side Length into the Formula:
[tex]\[
s = 23.62 \text{ feet}
\][/tex]
Calculate the area using the formula:
[tex]\[
A = \left( \frac{5}{4} \right) \cdot (23.62)^2 \cdot \left( \frac{1}{\tan(\pi/5)} \right)
\][/tex]
4. Computations (Intermediary steps are here for clarity):
[tex]\[
\left( \frac{5}{4} \right) \approx 1.25
\][/tex]
[tex]\[
s^2 = (23.62)^2 \approx 557.2644
\][/tex]
[tex]\[
\frac{1}{\tan(\pi/5)} \approx 1.3763819204711735
\][/tex]
Combining these, we get:
[tex]\[
A = 1.25 \times 557.2644 \times 1.3763819204711735 \approx 960.44
\][/tex]
5. Round to the Nearest Square Foot:
[tex]\[
\text{Area} \approx 960 \text{ square feet}
\][/tex]
Therefore, the area of the pool that needs to be covered is approximately [tex]\(960\)[/tex] square feet.
Hence, the correct answer is:
[tex]\[
\boxed{960 \, \text{ft}^2}
\][/tex]
