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. 45
B. 42
C. 50
D. 60

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Problem
We have a point P that is 25 m above a lake. From point P, the angle of elevation to a cloud is 30 degrees, and the angle of depression to the reflection of the cloud in the lake is 60 degrees. We need to find the height of the cloud above the lake's surface.

### Step 2: Draw the Diagram
Draw a diagram to visualize the problem:
- Let the height of the cloud above the lake be \( H \).
- The height of point P above the lake is 25 m.
- The angle of elevation to the cloud from P is 30 degrees.
- The angle of depression to the reflection of the cloud in the lake from P is 60 degrees.

### Step 3: Set Up the Right Triangles
1. For the angle of elevation (30 degrees):
- The height from point P to the cloud is \( H + 25 \) m.
- Let the horizontal distance from point P to the point directly below the cloud be \( PM \).

Using the tangent function:
\[ \tan(30^\circ) = \frac{H}{PM} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{H}{PM} \]
From this, we can express \( PM \):
\[ PM = H \sqrt{3} \]

2. For the angle of depression (60 degrees):
- The height from point P to the reflection of the cloud in the lake is \( H + 25 + 25 = H + 50 \) m.

Using the tangent function again:
\[ \tan(60^\circ) = \frac{H + 50}{PM} \]
We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{H + 50}{PM} \]
Substituting \( PM = H \sqrt{3} \):
\[ \sqrt{3} = \frac{H + 50}{H \sqrt{3}} \]

### Step 4: Solve for H
Cross-multiplying gives:
\[ \sqrt{3} \cdot H \sqrt{3} = H + 50 \]
This simplifies to:
\[ 3H = H + 50 \]
Rearranging gives:
\[ 3H - H = 50 \implies 2H = 50 \implies H = 25 \text{ m} \]

### Step 5: Find the Total Height of the Cloud
The total height of the cloud above the lake is:
\[ H + 25 = 25 + 25 = 50 \text{ m} \]

### Final Answer
The height of the cloud from the surface of the lake is **50 meters**.

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