From the top of a cliff, 200m high, the angle of depression of the top and bottom of a tower are observed to be `30^(@)` and `60^(@)`, find the height of the tower.

Answer

To solve the problem, we will use the concept of angles of depression and some basic trigonometry.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a cliff that is 200 meters high.
- From the top of the cliff, the angles of depression to the top and bottom of the tower are 30° and 60°, respectively.

2. **Draw the Diagram**:
- Draw a vertical line representing the cliff (200 m).
- Let point A be the top of the cliff, point B be the bottom of the tower, and point C be the top of the tower.
- The angle of depression to point C (top of the tower) is 30° and to point B (bottom of the tower) is 60°.

3. **Identify the Right Triangles**:
- From point A to point B, we have a right triangle where the height of the cliff is the opposite side and the horizontal distance from point A to point B is the adjacent side.
- Similarly, from point A to point C, we have another right triangle.

4. **Using Trigonometry for Triangle ACB**:
- For triangle ACB (where angle of depression to C is 30°):
\[ \tan(30°) = \frac{200}{x} \]
where \( x \) is the horizontal distance from point A to point C.
- We know that \( \tan(30°) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{200}{x} \]
- Rearranging gives:
\[ x = 200 \sqrt{3} \]

5. **Using Trigonometry for Triangle ADB**:
- For triangle ADB (where angle of depression to B is 60°):
\[ \tan(60°) = \frac{200}{x'} \]
where \( x' \) is the horizontal distance from point A to point B.
- We know that \( \tan(60°) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{200}{x'} \]
- Rearranging gives:
\[ x' = \frac{200}{\sqrt{3}} \]

6. **Finding the Height of the Tower**:
- The height of the tower \( H \) can be found by subtracting the height of point B from point C:
\[ H = (200 - \text{height at B}) \]
- The height at B can be calculated using the triangle ADB:
\[ \text{height at B} = 200 - H \]
- Using the horizontal distance \( x' \) from A to B:
\[ H = 200 - \left(200 - \frac{200}{\sqrt{3}}\right) \]
- Solving this gives:
\[ H = \frac{200}{\sqrt{3}} \text{ meters} \]

7. **Final Calculation**:
- To find the height of the tower, we need to calculate:
\[ H = 200 - \left(200 - \frac{200}{\sqrt{3}}\right) = \frac{200}{\sqrt{3}} \text{ meters} \]
- Therefore, the height of the tower is:
\[ H = 200 \left(1 - \frac{1}{\sqrt{3}}\right) = \frac{200(\sqrt{3} - 1)}{\sqrt{3}} \text{ meters} \]

### Final Answer:
The height of the tower is approximately \( 200(\sqrt{3} - 1) \) meters.