If the angle of elevation of a cloud from a point 10 metres above a lake is `30^(@)` and the angle of depression of its reflection in the lake is `60^(@)`. Find the height of the cloud from the surface of lake.

Answer

To find the height of the cloud from the surface of the lake, we can follow these steps:

### Step 1: Understand the Problem
We have a point 10 meters above the lake (let's call it point P). From this point, 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 C') is \(60^\circ\).

### Step 2: Draw a Diagram
Draw a diagram with:
- A horizontal line representing the surface of the lake.
- Point P above the lake, 10 meters high.
- Point C representing the cloud above point P.
- Point C' representing the reflection of the cloud in the lake.

### Step 3: Define Variables
Let:
- \(h\) = height of the cloud above the lake.
- The height of point C from the lake surface is \(h + 10\) meters (since point P is 10 meters above the lake).

### Step 4: Use Trigonometry for the Cloud
In triangle \(PCM\):
- The angle of elevation \( \angle CPM = 30^\circ\).
- The height from point P to the cloud is \(h\).
- The distance from point P to the base of the triangle (point M) is \(PM\).

Using the tangent function:
\[ \tan(30^\circ) = \frac{h}{PM} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{h}{PM} \implies PM = h \sqrt{3} \quad \text{(1)} \]

### Step 5: Use Trigonometry for the Reflection
In triangle \(PC'M\):
- The angle of depression \( \angle PC'M = 60^\circ\).
- The height from point P to the reflection C' is \(h + 10 + 10 = h + 20\).
- The distance from point P to the base of the triangle (point M) is still \(PM\).

Using the tangent function:
\[ \tan(60^\circ) = \frac{h + 20}{PM} \]
Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{h + 20}{PM} \implies PM = \frac{h + 20}{\sqrt{3}} \quad \text{(2)} \]

### Step 6: Equate the Two Expressions for PM
From equations (1) and (2):
\[ h \sqrt{3} = \frac{h + 20}{\sqrt{3}} \]

### Step 7: Solve for h
Cross-multiplying gives:
\[ h \sqrt{3} \cdot \sqrt{3} = h + 20 \implies 3h = h + 20 \]
\[ 3h - h = 20 \implies 2h = 20 \implies h = 10 \text{ meters} \]

### Step 8: Find the Height of the Cloud from the Lake
The total height of the cloud from the lake surface is:
\[ h + 10 = 10 + 10 = 20 \text{ meters} \]

### Final Answer
The height of the cloud from the surface of the lake is **20 meters**.

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