The cliff a mountain is 180 m high and the angles of depression of two ships on the either side of cliff are `30^(@)and60^(@)` .What is the distance between the tow ships ?

Options

A. 400 metres
B. `400sqrt(3)` metre
C. `145.68` metre
D. `39.6`metre

Answer

To solve the problem, we will use the concept of trigonometry, specifically the tangent function, which relates the angles of depression to the heights and distances involved.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a cliff (height = 180 m) and two ships on either side of the cliff.
- The angles of depression from the top of the cliff to the ships are 30° and 60°.

2. **Draw a Diagram**:
- Let point A be the top of the cliff, point D be the base of the cliff directly below A, and points B and C be the positions of the two ships.
- The height AD = 180 m.
- The angle of depression to ship B is 30° and to ship C is 60°.

3. **Identify the Right Triangles**:
- Triangle ABD for ship B and triangle ACD for ship C.
- In triangle ABD, angle ADB = 30°.
- In triangle ACD, angle ADC = 60°.

4. **Calculate Distance BD** (from A to B):
- Using the tangent function:
\[ \tan(30°) = \frac{AD}{BD} \]
- Substituting the known values:
\[ \tan(30°) = \frac{180}{BD} \]
- We know that \(\tan(30°) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{180}{BD} \]
- Rearranging gives:
\[ BD = 180 \sqrt{3} \]

5. **Calculate Distance CD** (from A to C):
- Using the tangent function again:
\[ \tan(60°) = \frac{AD}{CD} \]
- Substituting the known values:
\[ \tan(60°) = \frac{180}{CD} \]
- We know that \(\tan(60°) = \sqrt{3}\):
\[ \sqrt{3} = \frac{180}{CD} \]
- Rearranging gives:
\[ CD = \frac{180}{\sqrt{3}} = 60\sqrt{3} \]

6. **Calculate the Distance Between the Two Ships (BC)**:
- The distance between the two ships is given by:
\[ BC = BD + CD \]
- Substituting the values we found:
\[ BC = 180\sqrt{3} + 60\sqrt{3} = 240\sqrt{3} \]

7. **Approximate the Distance**:
- Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ BC \approx 240 \times 1.732 = 415.68 \text{ m} \]

### Final Answer:
The distance between the two ships is approximately **415.68 meters**.