From the top of a cliff 200 ft. high, the angles of depression of the top and bottom of a tower are observed to be `30^(@)` and `60^(@)` respectively. The height of the tower is____

Answer

To solve the problem, we need to find the height of the tower based on the given information about the angles of depression from the top of a 200 ft high cliff.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have a cliff that is 200 ft 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. **Setting Up the Diagram:**
- 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 height of the tower is represented as \( h \).
- The distance from the base of the cliff to the base of the tower is represented as \( d \).

3. **Using the Angle of Depression:**
- The angle of depression to the top of the tower (C) is 30°.
- The angle of depression to the bottom of the tower (B) is 60°.

4. **Applying Trigonometric Ratios:**
- For angle 30° (to point C):
\[ \tan(30°) = \frac{\text{Height of cliff - Height of tower}}{\text{Distance}} = \frac{200 - h}{d} \]
Since \( \tan(30°) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{200 - h}{d} \quad \text{(1)} \]

- For angle 60° (to point B):
\[ \tan(60°) = \frac{\text{Height of cliff}}{\text{Distance}} = \frac{200}{d} \]
Since \( \tan(60°) = \sqrt{3} \):
\[ \sqrt{3} = \frac{200}{d} \quad \text{(2)} \]

5. **Solving for Distance \( d \):**
- From equation (2):
\[ d = \frac{200}{\sqrt{3}} \quad \text{(3)} \]

6. **Substituting \( d \) into Equation (1):**
- Substitute \( d \) from equation (3) into equation (1):
\[ \frac{1}{\sqrt{3}} = \frac{200 - h}{\frac{200}{\sqrt{3}}} \]
- Cross-multiplying gives:
\[ 200 - h = \frac{200}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} = \frac{200}{3} \]

7. **Solving for Height \( h \):**
- Rearranging the equation:
\[ 200 - h = \frac{200}{3} \]
\[ h = 200 - \frac{200}{3} \]
- Converting 200 to a fraction:
\[ h = \frac{600}{3} - \frac{200}{3} = \frac{400}{3} \]
- Therefore, the height of the tower is:
\[ h = \frac{400}{3} \approx 133.33 \text{ ft} \]

### Final Answer:
The height of the tower is approximately **133.33 ft**.