From a light-house L two ships P and Q are observed in direction south-west and `15^(@)` east of south respectively. At the same time Q is observed from point P in south-east direction. If `LP=4 km`, the distance between the ships is

Options

A. 4 km
B. `4sqrt(3)` km
C. `5sqrt(3)` km
D. None of these

Answer

To solve the problem, we will break it down into steps and use trigonometry to find the distance between the two ships P and Q.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a lighthouse L from which two ships P and Q are observed. Ship P is in the southwest direction, and ship Q is in the direction of 15 degrees east of south. The distance from the lighthouse to ship P (LP) is given as 4 km.

2. **Drawing the Diagram**:
- Draw a coordinate system with L at the origin.
- Mark the direction of south and east.
- From L, draw a line towards ship P in the southwest direction (which is 45 degrees south of west).
- From L, draw another line towards ship Q at an angle of 15 degrees east of south.

3. **Identifying Angles**:
- The angle between the line to ship P and the south direction is 45 degrees (since southwest is 45 degrees from south).
- The angle between the line to ship Q and the south direction is 15 degrees.
- The angle between the lines to ships P and Q can be calculated as follows:
- The total angle from the south to ship P is 45 degrees.
- The angle from the south to ship Q is 15 degrees.
- Therefore, the angle between the two lines (angle PQL) is \( 45° + 15° = 60° \).

4. **Using Triangle Properties**:
- From the above angles, we can form triangle LQP.
- The angle at L (angle LQP) can be calculated as:
\[ \text{Angle LQP} = 180° - (60° + 90°) = 30° \]

5. **Applying the Law of Sines**:
- In triangle LQP, we know:
- \( LP = 4 \) km (the side opposite to angle Q).
- Angle LQP = 30°.
- Angle PQL = 60°.
- We can use the sine rule:
\[ \frac{PQ}{\sin(30°)} = \frac{LP}{\sin(60°)} \]
- Rearranging gives:
\[ PQ = LP \cdot \frac{\sin(30°)}{\sin(60°)} \]

6. **Calculating the Values**:
- We know:
- \( \sin(30°) = \frac{1}{2} \)
- \( \sin(60°) = \frac{\sqrt{3}}{2} \)
- Substituting these values:
\[ PQ = 4 \cdot \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = 4 \cdot \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} = 4\sqrt{3} \text{ km} \]

### Final Answer:
The distance between the ships P and Q is \( 4\sqrt{3} \) km.