From the top of a lighthouse. 100 m high, the angle of depression of two ships are 30° and 45°, if both ships are on same side find the distance between the ships ?

Options

A. A. 120 m
B. B. 180 m
C. C. 240 m
D. D. 360 m

Answer

To solve the problem step by step, we will use the concepts of trigonometry, specifically the tangent function, to find the distances from the lighthouse to each ship and then calculate the distance between the two ships.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a lighthouse that is 100 m high.
- The angle of depression to the first ship is 30° and to the second ship is 45°.
- We need to find the distance between the two ships.

2. **Draw a Diagram**:
- Let A be the top of the lighthouse, B the base of the lighthouse, C the position of the first ship, and D the position of the second ship.
- The height AB = 100 m.
- The angle of depression to ship C is 30°, and to ship D is 45°.

3. **Identify the Right Triangles**:
- Triangle ABC (larger triangle) has angle ACB = 30°.
- Triangle ABD (smaller triangle) has angle ADB = 45°.

4. **Use Trigonometric Ratios**:
- For triangle ABC:
\[ \tan(30°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} = \frac{100}{BD + DC} \]
- We know that \(\tan(30°) = \frac{1}{\sqrt{3}}\).
- So, we have:
\[ \frac{1}{\sqrt{3}} = \frac{100}{BD + DC} \]
- Rearranging gives:
\[ BD + DC = 100\sqrt{3} \quad \text{(Equation 1)} \]

5. **For Triangle ABD**:
- Using the same method:
\[ \tan(45°) = \frac{AB}{BD} = \frac{100}{BD} \]
- We know that \(\tan(45°) = 1\).
- So, we have:
\[ 1 = \frac{100}{BD} \]
- Rearranging gives:
\[ BD = 100 \quad \text{(Equation 2)} \]

6. **Substitute Equation 2 into Equation 1**:
- From Equation 1:
\[ 100 + DC = 100\sqrt{3} \]
- Solving for DC gives:
\[ DC = 100\sqrt{3} - 100 \]
- Factor out 100:
\[ DC = 100(\sqrt{3} - 1) \]

7. **Calculate the Value of DC**:
- Using \(\sqrt{3} \approx 1.732\):
\[ DC = 100(1.732 - 1) = 100(0.732) = 73.2 \text{ m} \]

8. **Conclusion**:
- The distance between the two ships (CD) is approximately 73.2 meters.