A transmitting antenna at te top of a tower has a height 64 m and the height of the receiving antenna is 100 m. The maximum distance between them for satisfactory communication in LOS mode will be {approximately}

Options

A. 45.5 km
B. 91 km
C. 136.5 km
D. 182.0 km

Explanation

To solve the problem of finding the maximum distance for satisfactory communication in Line of Sight (LOS) mode between a transmitting antenna and a receiving antenna, we can use the formula for the distance to the radio horizon. The formula is given by:

\[ d_m = \sqrt{2 \cdot R \cdot h_T} + \sqrt{2 \cdot R \cdot h_R} \]

Where:
- \( d_m \) = maximum distance for satisfactory communication
- \( R \) = radius of the Earth (approximately \( 6.4 \times 10^6 \) meters)
- \( h_T \) = height of the transmitting antenna (64 meters)
- \( h_R \) = height of the receiving antenna (100 meters)

### Step-by-Step Solution:

1. **Identify the given values**:
- Height of transmitting antenna, \( h_T = 64 \, \text{m} \)
- Height of receiving antenna, \( h_R = 100 \, \text{m} \)
- Radius of the Earth, \( R = 6.4 \times 10^6 \, \text{m} \)

2. **Calculate the first term**:
\[ \sqrt{2 \cdot R \cdot h_T} = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 64} \]
- Calculate \( 2 \cdot 6.4 \times 10^6 \cdot 64 \):
\[ 2 \cdot 6.4 \times 10^6 \cdot 64 = 819.2 \times 10^6 \]
- Now take the square root:
\[ \sqrt{819.2 \times 10^6} \approx 28.62 \, \text{km} \]

3. **Calculate the second term**:
\[ \sqrt{2 \cdot R \cdot h_R} = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 100} \]
- Calculate \( 2 \cdot 6.4 \times 10^6 \cdot 100 \):
\[ 2 \cdot 6.4 \times 10^6 \cdot 100 = 1280 \times 10^6 \]
- Now take the square root:
\[ \sqrt{1280 \times 10^6} \approx 35.77 \, \text{km} \]

4. **Add both terms to find the maximum distance**:
\[ d_m = 28.62 \, \text{km} + 35.77 \, \text{km} = 64.39 \, \text{km} \]

5. **Round the answer**:
The maximum distance for satisfactory communication in LOS mode is approximately:
\[ d_m \approx 64.40 \, \text{km} \]

### Final Answer:
The maximum distance between the transmitting and receiving antennas for satisfactory communication in LOS mode is approximately **64.40 km**.