If the perimeter of a sector of a circle of radius 5.2 cm is 16.4 cm, then the area of the sector is __________.

Options

A. `15.1 cm^(2)`
B. `15.5 cm^(2)`
C. `15.6 cm^(2)`
D. `15.9 cm^(2)`

Answer

To solve the problem step by step, we need to find the area of the sector given the perimeter of the sector and the radius of the circle.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the circle (r) = 5.2 cm
- Perimeter of the sector (P) = 16.4 cm

2. **Understand the Formula for the Perimeter of a Sector:**
The perimeter \( P \) of a sector of a circle can be expressed as:
\[ P = r + r + \text{Length of Arc} = 2r + \text{Length of Arc} \]
Here, the length of the arc can be expressed in terms of the angle \( \theta \) (in degrees) as:
\[ \text{Length of Arc} = \frac{\theta}{360} \times 2\pi r \]
Therefore, the formula for the perimeter becomes:
\[ P = 2r + \frac{\theta}{360} \times 2\pi r \]

3. **Substituting the Known Values:**
Substitute \( r = 5.2 \) cm and \( P = 16.4 \) cm into the perimeter formula:
\[ 16.4 = 2(5.2) + \frac{\theta}{360} \times 2\pi(5.2) \]
Simplifying this gives:
\[ 16.4 = 10.4 + \frac{\theta}{360} \times 10.4\pi \]

4. **Isolate the Arc Length Term:**
Rearranging the equation, we find:
\[ 16.4 - 10.4 = \frac{\theta}{360} \times 10.4\pi \]
\[ 6 = \frac{\theta}{360} \times 10.4\pi \]

5. **Solve for \( \theta \):**
Multiply both sides by \( 360 \) to isolate \( \theta \):
\[ 6 \times 360 = \theta \times 10.4\pi \]
\[ 2160 = \theta \times 10.4\pi \]
Therefore,
\[ \theta = \frac{2160}{10.4\pi} \]

6. **Calculate the Area of the Sector:**
The area \( A \) of the sector can be calculated using the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
Substituting \( r = 5.2 \) cm and the expression for \( \theta \):
\[ A = \frac{\frac{2160}{10.4\pi}}{360} \times \pi (5.2)^2 \]
Simplifying this further:
\[ A = \frac{2160 \times (5.2)^2}{360 \times 10.4} \]

7. **Calculate the Numerical Value:**
- Calculate \( (5.2)^2 = 27.04 \)
- Now substituting this value:
\[ A = \frac{2160 \times 27.04}{360 \times 10.4} \]
- Calculate \( 360 \times 10.4 = 3744 \)
- Now calculate:
\[ A = \frac{58378.4}{3744} \approx 15.6 \text{ cm}^2 \]

### Final Answer:
The area of the sector is \( 15.6 \text{ cm}^2 \).