A transmitting antenna at the top of a tower has height 32 m and height of the receiving antenna is 50 m. What is the maximum distance between them for satisfactory communication in line of sight (LOS) mode?

Options

A. 55.4 km
B. 45.5 km
C. 54.5km
D. 455 km

Explanation

To solve the problem of finding the maximum distance for satisfactory communication in line of sight (LOS) mode between a transmitting antenna on a tower and a receiving antenna, we can use the following steps:

### Step-by-Step Solution:

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

2. **Use the Formula for Maximum Distance**:
The formula for the maximum distance (D_max) in line of sight communication is given by:
\[ D_{max} = \sqrt{2 \cdot R \cdot H_T} + \sqrt{2 \cdot R \cdot H_R} \]
where R is the radius of the Earth, approximately \( R = 6.4 \times 10^6 \) m.

3. **Calculate the Contribution from the Transmitting Antenna**:
\[ D_T = \sqrt{2 \cdot R \cdot H_T} = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 32} \]
- Calculate \( 2 \cdot R \cdot H_T \):
\[ 2 \cdot 6.4 \times 10^6 \cdot 32 = 409600000 \quad \text{(m)} \]
- Now take the square root:
\[ D_T = \sqrt{409600000} \approx 202.4 \text{ m} \]

4. **Calculate the Contribution from the Receiving Antenna**:
\[ D_R = \sqrt{2 \cdot R \cdot H_R} = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 50} \]
- Calculate \( 2 \cdot R \cdot H_R \):
\[ 2 \cdot 6.4 \times 10^6 \cdot 50 = 640000000 \quad \text{(m)} \]
- Now take the square root:
\[ D_R = \sqrt{640000000} \approx 800 \text{ m} \]

5. **Calculate the Total Maximum Distance**:
\[ D_{max} = D_T + D_R = 202.4 \text{ m} + 800 \text{ m} = 1002.4 \text{ m} \]

6. **Convert to Kilometers**:
\[ D_{max} \approx 1.0024 \text{ km} \approx 1.0 \text{ km} \]

### Final Answer:
The maximum distance for satisfactory communication in line of sight (LOS) mode is approximately **1.0 km**.