The angle of elevation of the cloud at a point 2500 m high from the lake is `15^(@)` and the angle of depression of its reflection to the lake is `45^(@)`. Then, the height of cloud from the foot of lake is

Options

A. `2500sqrt(3)`m
B. 2500 m
C. `500 sqrt(3)`m
D. none of these

Answer

To solve the problem step-by-step, we will follow the given information and apply trigonometric concepts.

### Step 1: Understand the problem and draw a diagram
We have a lake, and a point 2500 m above the lake. The angle of elevation to the cloud from this point is 15 degrees, and the angle of depression to the reflection of the cloud in the lake is 45 degrees.

### Step 2: Define the variables
Let:
- \( h \) = height of the cloud above the lake
- \( x \) = horizontal distance from the point to the base of the cloud

### Step 3: Set up the equations using trigonometry
1. From the point 2500 m above the lake, the angle of elevation to the cloud is 15 degrees:
\[ \tan(15^\circ) = \frac{h - 2500}{x} \]
Rearranging gives:
\[ h - 2500 = x \cdot \tan(15^\circ) \quad \text{(1)} \]

2. The angle of depression to the reflection of the cloud in the lake is 45 degrees. The reflection is at a height of \( h + 2500 \) (since it goes down to the lake):
\[ \tan(45^\circ) = \frac{h + 2500}{x} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ h + 2500 = x \quad \text{(2)} \]

### Step 4: Solve the equations
From equation (2):
\[ x = h + 2500 \]
Substituting \( x \) in equation (1):
\[ h - 2500 = (h + 2500) \cdot \tan(15^\circ) \]
Now, we know \( \tan(15^\circ) = 2 - \sqrt{3} \):
\[ h - 2500 = (h + 2500)(2 - \sqrt{3}) \]
Expanding the right side:
\[ h - 2500 = (2 - \sqrt{3})h + (2 - \sqrt{3})2500 \]
Rearranging gives:
\[ h - (2 - \sqrt{3})h = 2500(2 - \sqrt{3}) + 2500 \]
Factoring out \( h \):
\[ h(1 - (2 - \sqrt{3})) = 2500(2 - \sqrt{3} + 1) \]
Simplifying:
\[ h(\sqrt{3} - 1) = 2500(3 - \sqrt{3}) \]
Thus, we find \( h \):
\[ h = \frac{2500(3 - \sqrt{3})}{\sqrt{3} - 1} \]

### Step 5: Calculate the height of the cloud from the foot of the lake
The total height of the cloud from the foot of the lake is:
\[ \text{Total height} = 2500 + h \]
Substituting \( h \):
\[ \text{Total height} = 2500 + \frac{2500(3 - \sqrt{3})}{\sqrt{3} - 1} \]
Factoring out \( 2500 \):
\[ \text{Total height} = 2500 \left(1 + \frac{3 - \sqrt{3}}{\sqrt{3} - 1}\right) \]
Simplifying the expression:
\[ = 2500 \left(\frac{(\sqrt{3} - 1) + (3 - \sqrt{3})}{\sqrt{3} - 1}\right) = 2500 \left(\frac{2}{\sqrt{3} - 1}\right) \]
Finally, we can express the total height:
\[ = 2500 \cdot \sqrt{3} \]

### Final Answer
The height of the cloud from the foot of the lake is:
\[ \text{Height} = 2500 \sqrt{3} \text{ meters} \]

---