The radius of a circle is 14 cm and the area of the sector is `102.7 cm^(2)`. Find the central angle of the sector.

Answer

To find the central angle of the sector when the radius of the circle is 14 cm and the area of the sector is 102.7 cm², we can use the formula for the area of a sector:

### Step-by-Step Solution:

1. **Write the formula for the area of a sector**:
\[ \text{Area of a sector} = \frac{\pi r^2 \theta}{360} \]
where \( r \) is the radius and \( \theta \) is the central angle in degrees.

2. **Substitute the known values into the formula**:
Given:
- Radius \( r = 14 \) cm
- Area of the sector = \( 102.7 \) cm²

Substitute these values into the formula:
\[ 102.7 = \frac{\pi \times (14)^2 \times \theta}{360} \]

3. **Use the value of \( \pi \)**:
We can use \( \pi \approx \frac{22}{7} \):
\[ 102.7 = \frac{\frac{22}{7} \times (14)^2 \times \theta}{360} \]

4. **Calculate \( 14^2 \)**:
\[ 14^2 = 196 \]
So, the equation becomes:
\[ 102.7 = \frac{\frac{22}{7} \times 196 \times \theta}{360} \]

5. **Simplify the equation**:
First, multiply \( \frac{22}{7} \) and \( 196 \):
\[ \frac{22 \times 196}{7} = \frac{4312}{7} \]
Now, substitute this back into the equation:
\[ 102.7 = \frac{4312 \theta}{2520} \]
(since \( 360 = 360 \div 7 = 2520 \))

6. **Cross-multiply to solve for \( \theta \)**:
\[ 102.7 \times 2520 = 4312 \theta \]
Calculate \( 102.7 \times 2520 \):
\[ 102.7 \times 2520 = 258804 \]
Now, we have:
\[ 258804 = 4312 \theta \]

7. **Solve for \( \theta \)**:
\[ \theta = \frac{258804}{4312} \]
Performing the division:
\[ \theta \approx 60.019 \]

8. **Round the answer**:
The central angle \( \theta \) can be approximated to:
\[ \theta \approx 60^\circ \]

### Final Answer:
The central angle of the sector is approximately \( 60^\circ \).