The angle of elevation of a cloud from a height h above the level of water in a lake is `alpha` and the angle of depressionof its image in the lake is `beta`. Find the height of the cloud above the surface of the lake:

Options

A. `(hsin(beta-alpha))/(sin(alpha+beta))`
B. `h sin alpha`
C. `(hsin(alpha+beta))/(sin(beta-alpha))`
D. none of these

Answer

To solve the problem, we need to find the height of the cloud above the surface of the lake given the height \( h \) above the lake, the angle of elevation \( \alpha \), and the angle of depression \( \beta \).

### Step-by-Step Solution:

1. **Understand the Setup**:
- Let \( C \) be the position of the cloud.
- Let \( A \) be the point on the surface of the lake directly below the cloud.
- Let \( B \) be the point from which the angle of elevation \( \alpha \) is measured, which is at height \( h \) above the lake.
- The angle of elevation \( \alpha \) is formed at point \( B \) towards point \( C \).
- The angle of depression \( \beta \) is formed at point \( B \) towards point \( A' \) (the image of the cloud in the lake).

2. **Define the Heights**:
- Let \( H \) be the height of the cloud above the lake surface.
- The height of point \( B \) above the lake is \( h \).
- Therefore, the vertical distance from point \( B \) to the cloud \( C \) is \( H - h \).

3. **Using Trigonometry for Angle of Elevation**:
- From point \( B \), using the angle of elevation \( \alpha \):
\[ \tan(\alpha) = \frac{H - h}{d} \]
where \( d \) is the horizontal distance from point \( B \) to point \( A \).

4. **Using Trigonometry for Angle of Depression**:
- The angle of depression \( \beta \) to the image \( A' \) in the lake gives:
\[ \tan(\beta) = \frac{h}{d} \]

5. **Express \( d \) from the Angle of Depression**:
- Rearranging the equation for \( \tan(\beta) \):
\[ d = \frac{h}{\tan(\beta)} \]

6. **Substituting \( d \) into the Elevation Equation**:
- Substitute \( d \) into the equation for \( \tan(\alpha) \):
\[ \tan(\alpha) = \frac{H - h}{\frac{h}{\tan(\beta)}} \]
- This simplifies to:
\[ \tan(\alpha) = \frac{(H - h) \tan(\beta)}{h} \]

7. **Rearranging to Find \( H \)**:
- Rearranging gives:
\[ H - h = \frac{h \tan(\alpha)}{\tan(\beta)} \]
- Therefore:
\[ H = h + \frac{h \tan(\alpha)}{\tan(\beta)} \]
- This can be further simplified to:
\[ H = h \left(1 + \frac{\tan(\alpha)}{\tan(\beta)}\right) \]

### Final Result:
The height of the cloud above the surface of the lake is:
\[ H = h \left(1 + \frac{\tan(\alpha)}{\tan(\beta)}\right) \]