If the angle of elevation of a cloud from a point P which is 25 m above a lake be `30^(@)` and the angle of depression of reflection of the cloud in the lake from P be `60^(@)`, then the height of the cloud (in meters) from the surface of the lake is

Options

A. 50
B. 60
C. 45
D. 42

Explanation

To solve the problem, we need to find the height of the cloud from the surface of the lake given the angles of elevation and depression from a point above the lake. Let's break down the solution step by step.

### Step 1: Understanding the Problem
We have a point P that is 25 m above the lake. From point P, the angle of elevation to the cloud (point C) is \(30^\circ\), and the angle of depression to the reflection of the cloud in the lake (point D) is \(60^\circ\). We need to find the height of the cloud above the lake.

### Step 2: Draw the Diagram
1. Draw a horizontal line to represent the surface of the lake.
2. Mark point P, which is 25 m above the lake.
3. Draw a line from P to the cloud (C) making an angle of \(30^\circ\) with the horizontal.
4. Draw a line from P down to the reflection of the cloud in the lake (D) making an angle of \(60^\circ\) with the horizontal.

### Step 3: Set Up the Triangles
- Let the height of the cloud above the lake be \(h\).
- The distance from point P vertically down to the lake is 25 m.
- Therefore, the height of the cloud from point P to point C is \(h + 25\).

### Step 4: Using Trigonometric Ratios
1. **For Triangle PCD (angle of elevation)**:
\[ \tan(30^\circ) = \frac{h}{d} \]
where \(d\) is the horizontal distance from P to the point directly below the cloud (point E).
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \implies h = d \cdot \frac{1}{\sqrt{3}} \tag{1} \]

2. **For Triangle PBD (angle of depression)**:
\[ \tan(60^\circ) = \frac{h + 25}{d} \]
\[ \tan(60^\circ) = \sqrt{3} \implies h + 25 = d \cdot \sqrt{3} \tag{2} \]

### Step 5: Solve the Equations
From equation (1):
\[ d = h \sqrt{3} \]
Substituting \(d\) in equation (2):
\[ h + 25 = (h \sqrt{3}) \cdot \sqrt{3} \]
\[ h + 25 = 3h \]
Rearranging gives:
\[ 3h - h = 25 \]
\[ 2h = 25 \implies h = \frac{25}{2} = 12.5 \text{ m} \]

### Step 6: Conclusion
The height of the cloud from the surface of the lake is \(12.5\) meters.

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