A community pool that is shaped like a regular pentagon needs a new cover for the winter months. The radius of the pool is 20.10 ft. The pool is 23.62 ft on each side.

To the nearest square foot, what is the area of the pool that needs to be covered?

A. 192 ft²
B. 960 ft²
C. 1,920 ft²
D. 3,842 ft²



Answer :

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]