If height of transmitting tower increases by `21%` then the area to be covered increases by

Options

A. A) `10%`
B. B) `21%`
C. C) `42%`
D. D) `84%`

Answer

To solve the problem, we need to understand the relationship between the height of the transmitting tower and the area it covers.

### Step-by-Step Solution:

1. **Understand the Relationship**: The maximum distance \( D_{max} \) that a transmitting tower can cover is given by the formula:
\[ D_{max} = \sqrt{2 r h_t} \]
where \( r \) is the radius of the Earth and \( h_t \) is the height of the transmitting tower.

2. **Square the Equation**: To relate the area covered to the height, we square both sides:
\[ D_{max}^2 = 2 r h_t \]
This shows that \( D_{max}^2 \) (which is proportional to the area covered) is directly proportional to the height \( h_t \).

3. **Proportionality**: Since \( D_{max}^2 \) is directly proportional to \( h_t \), we can express this as:
\[ D_{max}^2 \propto h_t \]
This means that if the height increases, the area covered will also increase in the same proportion.

4. **Calculate the Increase**: If the height of the tower increases by 21%, we can express this as:
\[ h_t' = h_t + 0.21 h_t = 1.21 h_t \]
Since the area covered is proportional to the square of the height, we can find the new area:
\[ D_{max}'^2 = 2 r h_t' = 2 r (1.21 h_t) = 1.21 \times (2 r h_t) = 1.21 D_{max}^2 \]
This means the new area covered is \( 1.21 \) times the original area.

5. **Percentage Increase in Area**: The increase in area can be calculated as:
\[ \text{Increase} = D_{max}'^2 - D_{max}^2 = 1.21 D_{max}^2 - D_{max}^2 = 0.21 D_{max}^2 \]
This corresponds to a 21% increase in the area covered.

6. **Conclusion**: Therefore, if the height of the transmitting tower increases by 21%, the area to be covered also increases by 21%.

### Final Answer:
The area to be covered increases by **21%** (Option B).

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