The angle of elevation of a stationary cloud from a point 2500m m above a lake is `15^(@)` and the angle of depression of its image in the lake is `45^(@).` The height the cloud above the lake level is

Options

A. A. 2500 m
B. B. `2500 sqrt3 m`
C. C.`5000 m `
D. D. `500 sqrt3 m`

Answer

To solve the problem step by step, we will use trigonometric principles related to angles of elevation and depression.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have a point \( B \) that is 2500 m above the lake.
- The angle of elevation to the cloud \( C \) from point \( B \) is \( 15^\circ \).
- The angle of depression to the image of the cloud in the lake is \( 45^\circ \).

2. **Setting Up the Diagram:**
- Let \( h \) be the height of the cloud \( C \) above the lake.
- The height of point \( B \) above the lake is 2500 m.
- Therefore, the total height from the lake to the cloud \( C \) is \( h + 2500 \).

3. **Using Triangle \( ABC \) for Angle of Elevation:**
- In triangle \( ABC \):
- \( \tan(15^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{x} \)
- Rearranging gives us:
\[ h = x \cdot \tan(15^\circ) \]

4. **Using Triangle \( ABE \) for Angle of Depression:**
- In triangle \( ABE \):
- \( \tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h + 2500}{x} \)
- Since \( \tan(45^\circ) = 1 \), we have:
\[ h + 2500 = x \]

5. **Substituting for \( x \):**
- From the second equation, we can express \( x \) in terms of \( h \):
\[ x = h + 2500 \]

6. **Equating the Two Expressions for \( x \):**
- Substitute \( x \) from the second equation into the first equation:
\[ h = (h + 2500) \cdot \tan(15^\circ) \]

7. **Solving for \( h \):**
- Expanding this gives:
\[ h = h \cdot \tan(15^\circ) + 2500 \cdot \tan(15^\circ) \]
- Rearranging gives:
\[ h - h \cdot \tan(15^\circ) = 2500 \cdot \tan(15^\circ) \]
- Factoring out \( h \):
\[ h(1 - \tan(15^\circ)) = 2500 \cdot \tan(15^\circ) \]
- Therefore:
\[ h = \frac{2500 \cdot \tan(15^\circ)}{1 - \tan(15^\circ)} \]

8. **Calculating \( \tan(15^\circ) \):**
- Using the known value:
\[ \tan(15^\circ) = 2 - \sqrt{3} \]
- Substitute this value into the equation for \( h \):
\[ h = \frac{2500 \cdot (2 - \sqrt{3})}{1 - (2 - \sqrt{3})} \]
- Simplifying gives:
\[ h = \frac{2500 \cdot (2 - \sqrt{3})}{\sqrt{3} - 1} \]

9. **Finding the Height of the Cloud Above the Lake:**
- The height of the cloud above the lake is:
\[ \text{Height of cloud} = h + 2500 \]
- Substitute \( h \) to find the final height.

### Final Answer:
The height of the cloud above the lake is \( 2500\sqrt{3} \) meters.