The angle of elevation of a stationary cloud from a point 25 m above a lake is `15^(@)` and the angle of depression of its image in the lake is `45^(@)`. The height of the cloud above the lake level is

Options

A. 25m
B. `25sqrt(3)m`
C. 50m
D. `50sqrt(3)m`

Explanation

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

### Step 1: Understand the Setup
We have a point 25 meters above the lake (let's call this point A). From point A, the angle of elevation to the cloud (point C) is 15 degrees. The angle of depression to the image of the cloud in the lake (point D) is 45 degrees.

### Step 2: Draw the Diagram
1. Draw a horizontal line representing the lake.
2. Mark point A, which is 25 meters above the lake.
3. Draw a vertical line down to point D (the image of the cloud in the lake).
4. Mark point C as the position of the cloud above the lake.

### Step 3: Identify the Angles
- The angle of elevation from A to C is 15 degrees.
- The angle of depression from A to D is 45 degrees.

### Step 4: Use Trigonometry for Angle of Depression
From point A, the angle of depression to point D is 45 degrees. Since the angle of depression is equal to the angle of elevation from point D to A, we can conclude that:
- The height from point A to point D (the image in the lake) is equal to the horizontal distance from A to D.

Let the horizontal distance from A to D be \( x \). Therefore, using the tangent function:
\[ \tan(45^\circ) = \frac{25}{x} \implies x = 25 \text{ meters} \]

### Step 5: Use Trigonometry for Angle of Elevation
Now, let's analyze the triangle formed by points A, C, and the horizontal line through A. The height from point A to point C is \( h - 25 \) (where \( h \) is the height of the cloud above the lake). The horizontal distance from A to C is also \( x \).

Using the tangent function for the angle of elevation:
\[ \tan(15^\circ) = \frac{h - 25}{x} \]
Substituting \( x = 25 \):
\[ \tan(15^\circ) = \frac{h - 25}{25} \]
Thus,
\[ h - 25 = 25 \tan(15^\circ) \]
\[ h = 25 + 25 \tan(15^\circ) \]

### Step 6: Calculate \( \tan(15^\circ) \)
Using the tangent value:
\[ \tan(15^\circ) \approx 0.2679 \]
Substituting this value:
\[ h = 25 + 25 \times 0.2679 \]
\[ h = 25 + 6.6975 \approx 31.6975 \text{ meters} \]

### Step 7: Conclusion
The height of the cloud above the lake level is approximately:
\[ h \approx 31.7 \text{ meters} \]