A transmitting antenna of height 20 m and the receiving antenna of height h are separated by a distance of 40km for satisfactory communication in line of sight mode. Then the value of h is

Options

A. 40m
B. 45m
C. 30m
D. 25m

Explanation

To find the height \( h \) of the receiving antenna for satisfactory communication in line of sight mode, we can use the formula for maximum distance \( D \) in line of sight communication:

\[ D_{\text{max}} = \sqrt{2 h_T R_e} + \sqrt{2 h_R R_e} \]

Where:
- \( D_{\text{max}} \) is the maximum distance for satisfactory communication,
- \( h_T \) is the height of the transmitting antenna,
- \( h_R \) is the height of the receiving antenna,
- \( R_e \) is the radius of the Earth.

### Step-by-step Solution:

1. **Identify the given values**:
- Height of the transmitting antenna \( h_T = 20 \) m
- Height of the receiving antenna \( h_R = h \) m (unknown)
- Distance between antennas \( D = 40 \) km = \( 40 \times 10^3 \) m
- Radius of the Earth \( R_e = 6.4 \times 10^6 \) m

2. **Set up the equation**:
Using the formula for maximum distance:
\[ 40 \times 10^3 = \sqrt{2 \cdot 20 \cdot 6.4 \times 10^6} + \sqrt{2 \cdot h \cdot 6.4 \times 10^6} \]

3. **Calculate the first term**:
Calculate \( \sqrt{2 \cdot 20 \cdot 6.4 \times 10^6} \):
\[ \sqrt{2 \cdot 20 \cdot 6.4 \times 10^6} = \sqrt{256 \times 10^6} = 16 \times 10^3 \text{ m} \]

4. **Substitute into the equation**:
Substitute the calculated value back into the equation:
\[ 40 \times 10^3 = 16 \times 10^3 + \sqrt{2 \cdot h \cdot 6.4 \times 10^6} \]

5. **Isolate the square root term**:
Rearranging gives:
\[ \sqrt{2 \cdot h \cdot 6.4 \times 10^6} = 40 \times 10^3 - 16 \times 10^3 = 24 \times 10^3 \]

6. **Square both sides**:
Squaring both sides to eliminate the square root:
\[ 2 \cdot h \cdot 6.4 \times 10^6 = (24 \times 10^3)^2 \]
\[ 2 \cdot h \cdot 6.4 \times 10^6 = 576 \times 10^6 \]

7. **Solve for \( h \)**:
\[ h = \frac{576 \times 10^6}{2 \cdot 6.4 \times 10^6} \]
\[ h = \frac{576}{12.8} = 45 \text{ m} \]

### Final Answer:
The height \( h \) of the receiving antenna is \( 45 \) m.

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