The sound of a thunder is heard 6 s after the lightning flash from a cloud at an angle of `60^(@)` to the horizontal. Calculate the height of cloud above the ground. The velocity of sound in air is 340 m/s.

Answer

To solve the problem of calculating the height of the cloud above the ground, we will follow these steps:

### Step 1: Understand the Problem
We know that the sound of thunder is heard 6 seconds after the lightning flash. The angle of the cloud with respect to the horizontal is 60 degrees. The speed of sound in air is given as 340 m/s.

### Step 2: Calculate the Distance Sound Travels
The distance traveled by sound can be calculated using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Substituting the values:
\[ \text{Distance} = 340 \, \text{m/s} \times 6 \, \text{s} = 2040 \, \text{m} \]

### Step 3: Relate the Height and Distance
In the right triangle formed by the height of the cloud (H) and the distance (D) from the point directly below the cloud to the observer, we can use trigonometric relationships. The angle given is 60 degrees. The relationship between the height (H) and the distance (D) can be expressed as:
\[ \tan(60^\circ) = \frac{H}{D} \]
From trigonometry, we know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus,
\[ H = D \cdot \tan(60^\circ) = D \cdot \sqrt{3} \]

### Step 4: Substitute the Distance into the Height Equation
Now we substitute the value of D (which is 2040 m) into the height equation:
\[ H = 2040 \, \text{m} \cdot \sqrt{3} \]

### Step 5: Calculate the Height
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ H = 2040 \, \text{m} \cdot 1.732 \approx 3535.68 \, \text{m} \]

### Final Answer
The height of the cloud above the ground is approximately **3535.68 meters**.

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