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 use the formula for the area of a sector of a circle. The formula is given by:

\[ \text{Area of the sector} = \frac{\theta}{360} \times \pi r^2 \]

Where:
- \(\theta\) is the angle of the sector in degrees,
- \(r\) is the radius (distance the light reaches),
- \(\pi\) is a constant approximately equal to \(3.14\).

### Step-by-step Solution:

1. **Identify the values**:
- The angle \(\theta = 80^\circ\)
- The radius \(r = 16.5 \text{ km}\)
- Use \(\pi \approx 3.14\)

2. **Substitute the values into the formula**:
\[ \text{Area} = \frac{80}{360} \times 3.14 \times (16.5)^2 \]

3. **Calculate \((16.5)^2\)**:
\[ (16.5)^2 = 272.25 \]

4. **Substitute \((16.5)^2\) back into the area formula**:
\[ \text{Area} = \frac{80}{360} \times 3.14 \times 272.25 \]

5. **Simplify \(\frac{80}{360}\)**:
\[ \frac{80}{360} = \frac{2}{9} \]

6. **Now substitute this back into the area calculation**:
\[ \text{Area} = \frac{2}{9} \times 3.14 \times 272.25 \]

7. **Calculate \(3.14 \times 272.25\)**:
\[ 3.14 \times 272.25 \approx 854.415 \]

8. **Now multiply by \(\frac{2}{9}\)**:
\[ \text{Area} = \frac{2}{9} \times 854.415 \approx 189.84 \text{ km}^2 \]

9. **Final result**:
The area of the sea over which the ships are warned is approximately \(189.84 \text{ km}^2\).