If we take 2 cases as:
1. antenna is at height 20m
2. antenna is at 25m above ground level then the difference in range between two cases in percentage in n%. Find n

Options

A. 0.12
B. 0.25
C. 0.35
D. 0.4

Answer

To solve the problem, we need to find the percentage difference in range between two antennas at different heights. The formula for the range \( R \) of an antenna is given by:

\[ R = \sqrt{2 R_E H} \]

where \( R_E \) is the radius of the Earth and \( H \) is the height of the antenna.

### Step-by-Step Solution:

1. **Identify the heights of the antennas**:
- For the first case, the height \( H_1 = 20 \, \text{m} \).
- For the second case, the height \( H_2 = 25 \, \text{m} \).

2. **Calculate the range for both cases**:
- For the first case:
\[ R_1 = \sqrt{2 R_E H_1} = \sqrt{2 R_E \cdot 20} \]
- For the second case:
\[ R_2 = \sqrt{2 R_E H_2} = \sqrt{2 R_E \cdot 25} \]

3. **Find the difference in range**:
\[ R_2 - R_1 = \sqrt{2 R_E \cdot 25} - \sqrt{2 R_E \cdot 20} \]

4. **Factor out the common term**:
\[ R_2 - R_1 = \sqrt{2 R_E} \left( \sqrt{25} - \sqrt{20} \right) \]

5. **Simplify the square roots**:
\[ R_2 - R_1 = \sqrt{2 R_E} \left( 5 - \sqrt{20} \right) \]

6. **Calculate the percentage difference**:
The percentage difference in range is given by:
\[ \text{Percentage Difference} = \frac{R_2 - R_1}{R_1} \times 100 \]

7. **Substituting the values**:
\[ \text{Percentage Difference} = \frac{\sqrt{2 R_E} \left( 5 - \sqrt{20} \right)}{\sqrt{2 R_E} \sqrt{20}} \times 100 \]
The \( \sqrt{2 R_E} \) cancels out:
\[ \text{Percentage Difference} = \frac{5 - \sqrt{20}}{\sqrt{20}} \times 100 \]

8. **Calculate \( \sqrt{20} \)**:
\[ \sqrt{20} = 4.47 \]
Thus,
\[ \text{Percentage Difference} = \frac{5 - 4.47}{4.47} \times 100 \]

9. **Final Calculation**:
\[ \text{Percentage Difference} = \frac{0.53}{4.47} \times 100 \approx 11.85\% \]

10. **Conclusion**:
Therefore, the value of \( n \) is approximately \( 12 \).