A Television transmitting antenna is 84 m tall. How much service area can it cover, if the receiving antenna is at the ground level? (Given \( \pi = \displaystyle\frac{22}{7} \))
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 3379\,\mathrm{km^{2}} \)
Television signals travel in straight lines (line-of-sight or space-wave propagation), so an antenna of height \( h \) reaches only up to the point where its line of sight touches the curved earth. With \( R \) the radius of the earth, this range is
\( \displaystyle d = \sqrt{2hR} \)
The service area is the circle of radius \( d \) around the antenna:
\( \displaystyle A = \pi d^{2} = 2\pi h R \)
\( h = 84\,\mathrm{m} = 0.084\,\mathrm{km} \)
\( R = 6400\,\mathrm{km} \)
\( \displaystyle 2hR = 2 \times 0.084 \times 6400 = 1075.2\,\mathrm{km^{2}} \)
\( \displaystyle A = \frac{22}{7} \times 1075.2 \approx 3379\,\mathrm{km^{2}} \)
Note: This question is from the pre-rationalised NCERT book (2022-23). The topic was removed from the latest NCERT book (2026-27), and it also appears to be out of the CUET 2026 syllabus.
NCERT (Pre-rationalised edition, 2022-23):
Space Wave – “… If the transmitting antenna is at a height hT, then you can show that the distance to the horizon dT is given as \( d_T = \sqrt{2Rh_T} \), where R is the radius of the earth (approximately 6400 km).”