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 solve the problem of finding the minimum distance between a transmitting antenna and a receiving antenna for satisfactory Line of Sight (LOS) communication, we can use the formula for the maximum distance of communication based on the heights of the antennas.

### Step-by-Step Solution:

1. **Identify the Heights of the Antennas**:
- Height of transmitting antenna (H_t) = 32 m
- Height of receiving antenna (H_r) = 50 m

2. **Use the Formula for Maximum Distance**:
The formula for the maximum distance (D) for satisfactory LOS communication is given by:
\[ D = \sqrt{2H_t \cdot R_e} + \sqrt{2H_r \cdot R_e} \]
where \( R_e \) is the radius of the Earth, approximately \( 6.4 \times 10^6 \) m.

3. **Calculate Each Component**:
- First, calculate \( \sqrt{2H_t \cdot R_e} \):
\[ \sqrt{2 \cdot 32 \cdot (6.4 \times 10^6)} = \sqrt{64 \cdot 6.4 \times 10^6} = \sqrt{409600000} \approx 640 \text{ m} \]
- Next, calculate \( \sqrt{2H_r \cdot R_e} \):
\[ \sqrt{2 \cdot 50 \cdot (6.4 \times 10^6)} = \sqrt{100 \cdot 6.4 \times 10^6} = \sqrt{640000000} \approx 800 \text{ m} \]

4. **Add the Distances**:
Now, add the two components to find the total distance:
\[ D = 640 + 800 = 1440 \text{ m} \]

5. **Convert to Kilometers**:
Since the question might require the answer in kilometers, convert meters to kilometers:
\[ D = \frac{1440}{1000} = 1.44 \text{ km} \]

### Final Answer:
The minimum distance between the transmitting and receiving antennas for satisfactory LOS communication is approximately **1.44 km**.