The angle of elevation of a cloud from a point 250 m above a lake is `15^(@)` and angle of depression of its reflection in lake is `45^(@)`. The height of the cloud is

Options

A. `250 sqrt(3) m`
B. 250 m
C. `( 250 ) /( sqrt(3)) m `
D. None of these

Answer

To find the height of the cloud above the lake, we can break down the problem step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a point \( P \) which is 250 m above the lake.
- The angle of elevation to the cloud \( C \) from point \( P \) is \( 15^\circ \).
- The angle of depression to the reflection of the cloud \( C' \) in the lake is \( 45^\circ \).

2. **Setting Up the Diagram**:
- Let the height of the cloud \( C \) above the lake be \( h \).
- The height of point \( P \) above the lake is 250 m.
- The distance from point \( P \) to the vertical line below the cloud is \( d \).

3. **Using the Angle of Elevation**:
- From point \( P \), the angle of elevation to the cloud \( C \) gives us the equation:
\[ \tan(15^\circ) = \frac{h - 250}{d} \]
- Rearranging gives:
\[ h - 250 = d \cdot \tan(15^\circ) \quad \text{(1)} \]

4. **Using the Angle of Depression**:
- The angle of depression to the reflection \( C' \) gives us:
\[ \tan(45^\circ) = \frac{h + 250}{d} \]
- Since \( \tan(45^\circ) = 1 \), we have:
\[ h + 250 = d \quad \text{(2)} \]

5. **Substituting Equation (2) into Equation (1)**:
- From equation (2), we can express \( d \) in terms of \( h \):
\[ d = h + 250 \]
- Substitute this into equation (1):
\[ h - 250 = (h + 250) \cdot \tan(15^\circ) \]

6. **Expanding and Rearranging**:
- Expanding gives:
\[ h - 250 = h \cdot \tan(15^\circ) + 250 \cdot \tan(15^\circ) \]
- Rearranging leads to:
\[ h - h \cdot \tan(15^\circ) = 250 + 250 \cdot \tan(15^\circ) \]
- Factor out \( h \):
\[ h(1 - \tan(15^\circ)) = 250 + 250 \cdot \tan(15^\circ) \]

7. **Solving for \( h \)**:
- Thus, we have:
\[ h = \frac{250 + 250 \cdot \tan(15^\circ)}{1 - \tan(15^\circ)} \]

8. **Calculating Values**:
- Calculate \( \tan(15^\circ) \):
\[ \tan(15^\circ) \approx 0.2679 \]
- Substitute this value into the equation:
\[ h = \frac{250 + 250 \cdot 0.2679}{1 - 0.2679} \]
- Simplifying gives:
\[ h = \frac{250 + 66.975}{0.7321} \approx \frac{316.975}{0.7321} \approx 432.3 \text{ m} \]

9. **Final Height of the Cloud**:
- Therefore, the height of the cloud above the lake is approximately \( 432.3 \) m.