The angle of elevation of a cloud from a point 60m above a lake is `30^@` and the angle of depression of the reflection of cloud in the lake is `60^@`. Find the height of the cloud.

Answer

To find the height of the cloud above the lake, we will follow these steps:

### Step 1: Understand the problem and draw a diagram
We have a point \( O \) which is 60 meters above the lake. From this point, the angle of elevation to the cloud is \( 30^\circ \) and the angle of depression to the reflection of the cloud in the lake is \( 60^\circ \).

### Step 2: Define the variables
Let:
- \( H \) = height of the cloud above the lake
- \( OA = 60 \) m (height of point \( O \) above the lake)
- \( OB = OA + H = 60 + H \) (height of the cloud above the lake)
- \( OD \) = horizontal distance from point \( O \) to the point directly below the cloud.

### Step 3: Use the angle of elevation
From point \( O \), using the angle of elevation \( 30^\circ \):
\[ \tan(30^\circ) = \frac{H}{OD} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{H}{OD} \]
This gives us:
\[ OD = \sqrt{3}H \tag{1} \]

### Step 4: Use the angle of depression
From point \( O \), using the angle of depression \( 60^\circ \) to the reflection of the cloud:
\[ \tan(60^\circ) = \frac{OA + H}{OD} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{60 + H}{OD} \]
Substituting \( OD \) from equation (1):
\[ \sqrt{3} = \frac{60 + H}{\sqrt{3}H} \]
Cross-multiplying gives:
\[ 3H = 60 + H \]

### Step 5: Solve for \( H \)
Rearranging the equation:
\[ 3H - H = 60 \]
\[ 2H = 60 \]
\[ H = 30 \text{ m} \]

### Step 6: Find the total height of the cloud above the lake
The total height of the cloud above the lake is:
\[ AB = OA + H = 60 + 30 = 90 \text{ m} \]

### Final Answer
The height of the cloud above the lake is **90 meters**.

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