Question

A TV transmission tower antenna is at a height of 20 m. Suppose that the receiving antenna is at. (i) ground level (ii) a height of 5 m. The increase in antenna range in case (ii) relative to case (i) is n%. The value of n, to the nearest integer, is __________.

Hint:

Radio horizon range increases with the square root of the antenna height.

Answer 1

Correct Answer: 50

Step 1: The range of communication is given by \(d = \sqrt{2Rh_t} + \sqrt{2Rh_r}\).
Step 2: Case (i): \(h_r = 0\). \[ d_1 = \sqrt{2R(20)} + 0 = \sqrt{40R} \]
Step 3: Case (ii): \(h_r = 5\) m. \[ d_2 = \sqrt{2R(20)} + \sqrt{2R(5)} = \sqrt{40R} + \sqrt{10R} \] \[ d_2 = 2\sqrt{10R} + \sqrt{10R} = 3\sqrt{10R} \] Note that \(d_1 = \sqrt{4 \times 10R} = 2\sqrt{10R}\).
Step 4: Calculate percentage increase. \[ \text{Increase} = \frac{d_2 - d_1}{d_1} \times 100 = \frac{3\sqrt{10R} - 2\sqrt{10R}}{2\sqrt{10R}} \times 100 = \frac{1}{2} \times 100 = 50% \]