A transmitting antenna is at a height of 20m and receiving antenna is at a height of 80m. The maximum distance between satisfactory communication is them for

Options

A. 16 km
B. 32km
C. 48 km
D. 96 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) = 20 m
- Height of the receiving antenna (h_r) = 80 m

2. **Convert Heights to Kilometers**:
- Convert the heights from meters to kilometers:
- \( h_t = 20 \, \text{m} = 20 \times 10^{-3} \, \text{km} = 0.02 \, \text{km} \)
- \( h_r = 80 \, \text{m} = 80 \times 10^{-3} \, \text{km} = 0.08 \, \text{km} \)

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

4. **Calculate Each Term**:
- Calculate the first term:
\[ \sqrt{2 \cdot 6400 \cdot 0.02} = \sqrt{256} = 16 \, \text{km} \]
- Calculate the second term:
\[ \sqrt{2 \cdot 6400 \cdot 0.08} = \sqrt{1024} = 32 \, \text{km} \]

5. **Add the Two Distances**:
- Combine the distances calculated:
\[ d_{\text{max}} = 16 \, \text{km} + 32 \, \text{km} = 48 \, \text{km} \]

6. **Conclusion**:
- The maximum distance for satisfactory communication between the transmitting and receiving antennas is **48 km**.

### Final Answer:
The maximum distance between the transmitting antenna and the receiving antenna is **48 km**.

---