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

Let \( R \) be the radius of the Earth, and \( h \) be the height of the antenna. The distance \( d \) between the two antennas is given by: \[ d = \sqrt{2Rh} \] We are given that \( d = 4 \, \text{km} \), and the radius of the Earth \( R = 6400 \, \text{km} \). Substituting these values into the equation: \[ 4 = \sqrt{2 \times 6400 \times h} \] Squaring both sides: \[ 16 = 2 \times 6400 \times h \] \[ h = \frac{16}{12800} = 1 \, \text{m} \] Now, using the formula for the signal range: \[ x = \frac{500}{4} = 125 \] Thus, the value of \( x \) is 125.