A transmitting antenna is a height of 40 m and the receiving antenna is at a height of 60 m The maximum distance between them for satisfactory communication is nearly

Options

A. A) `22.5` km
B. B) `27.5` km
C. C) 50 km
D. D) 25 km

Answer

To solve the problem of finding the maximum distance for satisfactory communication between a transmitting antenna and a receiving antenna, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Heights of the Antennas:**
- Height of the transmitting antenna, \( h_t = 40 \, \text{m} \)
- Height of the receiving antenna, \( h_r = 60 \, \text{m} \)

2. **Convert Heights to Kilometers:**
- Convert the height of the transmitting antenna from meters to kilometers:
\[ h_t = 40 \, \text{m} = \frac{40}{1000} \, \text{km} = 0.04 \, \text{km} \]
- Convert the height of the receiving antenna from meters to kilometers:
\[ h_r = 60 \, \text{m} = \frac{60}{1000} \, \text{km} = 0.06 \, \text{km} \]

3. **Use the Formula for Maximum Distance:**
- The formula for the maximum distance \( D_{max} \) for satisfactory communication is given by:
\[ D_{max} = \sqrt{2 r h_t} + \sqrt{2 r h_r} \]
- Where \( r \) is the radius of the Earth, approximately \( r = 6400 \, \text{km} \).

4. **Substitute the Values into the Formula:**
- Substitute \( r \), \( h_t \), and \( h_r \) into the formula:
\[ D_{max} = \sqrt{2 \times 6400 \times 0.04} + \sqrt{2 \times 6400 \times 0.06} \]

5. **Calculate Each Term:**
- Calculate the first term:
\[ \sqrt{2 \times 6400 \times 0.04} = \sqrt{512} \approx 22.63 \, \text{km} \]
- Calculate the second term:
\[ \sqrt{2 \times 6400 \times 0.06} = \sqrt{768} \approx 27.71 \, \text{km} \]

6. **Add the Two Results:**
- Combine the results to find \( D_{max} \):
\[ D_{max} \approx 22.63 + 27.71 \approx 50.34 \, \text{km} \]

7. **Final Result:**
- Therefore, the maximum distance for satisfactory communication is approximately:
\[ D_{max} \approx 50 \, \text{km} \]

### Conclusion:
The maximum distance between the transmitting and receiving antennas for satisfactory communication is nearly **50 kilometers**.

---