To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle `80^@`to a distance of 16.5 km. Find the area of the sea over which the ships are warned.

Explanation

To find the area of the sea over which the ships are warned by the lighthouse, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the parameters**:
- The angle of the sector (θ) = 80 degrees
- The radius (r) = 16.5 km

2. **Formula for the area of a sector**:
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\theta}{360^\circ} \times \pi r^2 \]

3. **Substitute the values into the formula**:
- Substitute \( \theta = 80^\circ \) and \( r = 16.5 \) km into the formula:
\[ A = \frac{80}{360} \times \pi \times (16.5)^2 \]

4. **Simplify the fraction**:
- Simplifying \( \frac{80}{360} \):
\[ \frac{80}{360} = \frac{2}{9} \]

5. **Calculate \( (16.5)^2 \)**:
- Calculate \( 16.5 \times 16.5 \):
\[ (16.5)^2 = 272.25 \]

6. **Substitute \( \pi \)**:
- Use \( \pi \approx \frac{22}{7} \):
\[ A = \frac{2}{9} \times \frac{22}{7} \times 272.25 \]

7. **Calculate the area**:
- First, calculate \( \frac{2 \times 22}{9 \times 7} \):
\[ \frac{44}{63} \]
- Now multiply by \( 272.25 \):
\[ A = \frac{44 \times 272.25}{63} \]

8. **Perform the multiplication**:
- Calculate \( 44 \times 272.25 \):
\[ 44 \times 272.25 = 11979 \]

9. **Divide by 63**:
- Now divide \( 11979 \) by \( 63 \):
\[ A \approx 190.14 \text{ km}^2 \]

10. **Final Answer**:
The area of the sea over which the ships are warned is approximately \( 190.14 \text{ km}^2 \).