Question

The height of a television transmitting antenna is 70 m. If the receiving antenna is at the ground level, then the service area covered by the transmitting antenna is nearly (Radius of the earth = 6400 km)

• A. $2236\times10^{6}$ m$^{2}$
• B. $1408\times10^{6}$ m$^{2}$
• C. $3348\times10^{6}$ m$^{2}$
• D. $2816\times10^{6}$ m$^{2}$

Hint:

Formula $d = \sqrt{2 h R}$ assumes receiving antenna at ground; for height $h_r$, add $\sqrt{2 h_r R}$. Use SI units: km to m. Approximate $\sqrt{}$ and $\pi$ for quick computation.

Answer 1

The Correct Option is D

1. The service area for a transmitting antenna is the area within the line-of-sight distance, approximated for a curved Earth.
2. The maximum distance $d$ to the horizon is $d = \sqrt{2 h R}$, where $h$ is antenna height and $R$ is Earth's radius.
3. Given $h = 70$ m, $R = 6400$ km = $6.4 \times 10^6$ m.
4. Calculate $d = \sqrt{2 \times 70 \times 6.4 \times 10^6} = \sqrt{8.96 \times 10^8} \approx 2.993 \times 10^4$ m.
5. The service area is a circle: $A = \pi d^2 \approx 3.14 \times (2.993 \times 10^4)^2 \approx 2.816 \times 10^9$ m$^2$ = $2816 \times 10^6$ m$^2$.
6. Therefore, the correct option is (4) $2816\times10^{6}$ m$^{2}$.