If the angle of elevation of a cloud from a point 200m above a lake is `30^(@)` and the angle of depression of its reflection in the lake is `60^(@)` .Then the height of the cloud above the lake is :

Options

A. 100 m
B. 200 m
C. 300 m
D. 400 m

Answer

To solve the problem, we will use trigonometric principles involving angles of elevation and depression. Let's break down the solution step by step.

### Step 1: Understand the scenario
We have a point \( P \) that is 200 m above the lake. From point \( P \), the angle of elevation to the cloud \( C \) is \( 30^\circ \) and the angle of depression to the reflection of the cloud \( C' \) in the lake is \( 60^\circ \).

### Step 2: Draw a diagram
1. Draw a horizontal line to represent the lake.
2. Mark point \( P \) 200 m above the lake.
3. Draw the cloud \( C \) above point \( P \) and the reflection \( C' \) below the lake.
4. Mark the angles: \( \angle CPB = 30^\circ \) (elevation) and \( \angle PBC' = 60^\circ \) (depression).

### Step 3: Set up the problem
Let \( h \) be the height of the cloud \( C \) above the lake. Therefore, the height of the cloud from point \( P \) is \( h - 200 \) m.

### Step 4: Use the tangent function for angle of elevation
From point \( P \):
\[ \tan(30^\circ) = \frac{h - 200}{d} \]
Where \( d \) is the horizontal distance from point \( P \) to the point directly below the cloud.

Using \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h - 200}{d} \quad \text{(1)} \]

### Step 5: Use the tangent function for angle of depression
From point \( P \):
\[ \tan(60^\circ) = \frac{200 + h}{d} \]
Using \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{200 + h}{d} \quad \text{(2)} \]

### Step 6: Solve equations (1) and (2)
From equation (1):
\[ d = \sqrt{3}(h - 200) \quad \text{(3)} \]

From equation (2):
\[ d = \frac{200 + h}{\sqrt{3}} \quad \text{(4)} \]

### Step 7: Set equations (3) and (4) equal to each other
\[ \sqrt{3}(h - 200) = \frac{200 + h}{\sqrt{3}} \]

### Step 8: Cross-multiply to eliminate the fraction
\[ 3(h - 200) = 200 + h \]
Expanding gives:
\[ 3h - 600 = 200 + h \]

### Step 9: Rearrange the equation
\[ 3h - h = 200 + 600 \]
\[ 2h = 800 \]
\[ h = 400 \]

### Final Answer
The height of the cloud above the lake is **400 meters**.