The angle of elevation of a cloud from a point 10 meters above the surface of a lake is `30^(@)` and the angle of depression of its reflection from that point is `60^(@)`. Then the height of the could above the lake is

Options

A. 20 meters
B. `20sqrt3` meters
C. `(20(sqrt3-1))/(sqrt3)` meters
D. `20(sqrt3-1)` meters

Answer

To solve the problem, we will use trigonometric concepts involving angles of elevation and depression.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a point \( O \) which is 10 meters above the surface of the lake.
- The angle of elevation to the cloud from point \( O \) is \( 30^\circ \).
- The angle of depression to the reflection of the cloud in the lake from point \( O \) is \( 60^\circ \).

2. **Setting Up the Diagram**:
- Let the height of the cloud above the lake be \( h \).
- The distance from point \( O \) to the point directly below the cloud on the lake's surface is denoted as \( d \).
- The reflection of the cloud in the lake is at a height of \( h \) below the lake's surface, which is \( -h \).

3. **Using the Angle of Elevation**:
- From point \( O \) (10 m above the lake), the angle of elevation to the cloud is \( 30^\circ \).
- Using the tangent function:
\[ \tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h - 10}{d} \]
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h - 10}{d} \implies d = \sqrt{3}(h - 10) \]

4. **Using the Angle of Depression**:
- The angle of depression to the reflection of the cloud is \( 60^\circ \).
- The height from point \( O \) to the reflection (which is \( 10 + h \)) can be expressed as:
\[ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{10 + h}{d} \]
- Since \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ \sqrt{3} = \frac{10 + h}{d} \implies d = \frac{10 + h}{\sqrt{3}} \]

5. **Equating the Two Expressions for \( d \)**:
- From the two equations for \( d \):
\[ \sqrt{3}(h - 10) = \frac{10 + h}{\sqrt{3}} \]
- Cross-multiplying gives:
\[ 3(h - 10) = 10 + h \]
- Expanding and rearranging:
\[ 3h - 30 = 10 + h \implies 3h - h = 10 + 30 \implies 2h = 40 \implies h = 20 \]

6. **Finding the Height of the Cloud Above the Lake**:
- The height of the cloud above the lake is \( h = 20 \) meters.

### Final Answer:
The height of the cloud above the lake is **20 meters**.