A transmitting antenna at the top of tower has a height 32m and height of receiving antenna is 50m. Minimum distance between them for satisfactory LOS mode of communication is

Options

A. 40km
B. 45km
C. 60km
D. 100km

Answer

To find the minimum distance between a transmitting antenna and a receiving antenna for satisfactory Line of Sight (LOS) mode of communication, we can use the following formula:

\[ D = \sqrt{2R \cdot H_T} + \sqrt{2R \cdot H_R} \]

Where:
- \( D \) = minimum distance between the antennas
- \( R \) = radius of the Earth (approximately \( 6.4 \times 10^6 \) meters)
- \( H_T \) = height of the transmitting antenna
- \( H_R \) = height of the receiving antenna

### Step-by-Step Solution:

1. **Identify the given values:**
- Height of the transmitting antenna, \( H_T = 32 \) m
- Height of the receiving antenna, \( H_R = 50 \) m
- Radius of the Earth, \( R = 6.4 \times 10^6 \) m

2. **Substitute the values into the formula:**
\[ D = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 32} + \sqrt{2 \cdot (6.4 \times 10^6) \cdot 50} \]

3. **Calculate each term separately:**
- For the transmitting antenna:
\[ D_T = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 32} \]
\[ D_T = \sqrt{409600000} \approx 640 \text{ m} \]

- For the receiving antenna:
\[ D_R = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 50} \]
\[ D_R = \sqrt{640000000} \approx 800 \text{ m} \]

4. **Add the two distances together:**
\[ D = D_T + D_R = 640 \text{ m} + 800 \text{ m} = 1440 \text{ m} \]

5. **Convert the distance from meters to kilometers:**
\[ D = \frac{1440}{1000} = 1.44 \text{ km} \]

6. **Final Calculation:**
To find the satisfactory LOS distance, we can also express it in terms of square roots:
\[ D = \sqrt{2R(H_T + H_R)} = \sqrt{2 \cdot 6.4 \times 10^6 \cdot (32 + 50)} \]
\[ D = \sqrt{2 \cdot 6.4 \times 10^6 \cdot 82} \approx 45 \text{ km} \]

### Final Answer:
The minimum distance between the transmitting and receiving antennas for satisfactory LOS mode of communication is approximately **45 km**.