Question

A transmitting antenna is kept on the surface of the earth. The minimum height of receiving antenna required to receive the signal in line of sight at 4 km distance from it is \( x \times 10^{-2} \) m. The value of \( x \) is:

• A. 125
• B. 12.5
• C. 1250
• D. 1.25

Hint:

For problems involving the height of an antenna and the line of sight, use the relation \( d = \sqrt{2Rh} \) to calculate the minimum height for the signal to be received.

Answer 1

The Correct Option is A

To solve this problem, we need to determine the minimum height of the receiving antenna required to maintain a line of sight with the transmitting antenna located at a distance of 4 km on the earth's surface.

1. The line of sight distance (\(d\)) between two antennas can be calculated using the formula: \(d = \sqrt{2 \cdot R \cdot h_t} + \sqrt{2 \cdot R \cdot h_r}\) Where:
   • \(R\) is the radius of the Earth (approximately \(6371 \text{ km} = 6.371 \times 10^6 \text{ m}\)).
   • \(h_t\) is the height of the transmitting antenna.
   • \(h_r\) is the height of the receiving antenna.
2. In our scenario, the transmitting antenna is at the surface, so \(h_t = 0\) and the line of sight distance is given as 4 km (or 4000 m): \(4000 = \sqrt{2 \cdot 6.371 \times 10^6 \cdot 0} + \sqrt{2 \cdot 6.371 \times 10^6 \cdot h_r}\)
3. Since \(\sqrt{2 \cdot 6.371 \times 10^6 \cdot 0} = 0\), the formula simplifies to: \(4000 = \sqrt{2 \cdot 6.371 \times 10^6 \cdot h_r}\)
4. Squaring both sides to remove the square root: \(4000^2 = 2 \cdot 6.371 \times 10^6 \cdot h_r\)
5. Solving for \(h_r\): \(h_r = \frac{4000^2}{2 \cdot 6.371 \times 10^6}\) \(h_r = \frac{16000000}{12,742,000} \approx 1.2559 \text{ m}\)
6. Finally, since the required height of the receiving antenna is \( x \times 10^{-2} \) m, we identify: \(h_r = 125.59 \times 10^{-2} \text{ m}\), giving us \( x \approx 125 \).

Therefore, the value of \( x \) is 125.