Question:
A TV transmitting antenna is 81 m tall. Service area covered, if the receiving antenna is at the ground level, will be about:
A TV transmitting antenna is 81 m tall. Service area covered, if the receiving antenna is at the ground level, will be about:
Show Hint
For antenna horizon problems, memorize the final shortcut:
\[
A = 2\pi R_e h
\]
It directly gives area without computing distance separately.
Updated On: Jun 8, 2026
- \( 3257~\text{km}^{2} \)
- \( 4250~\text{km}^{2} \)
- \( 2500~\text{km}^{2} \)
- \( 1500~\text{km}^{2} \)
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The Correct Option is A
Solution and Explanation
Concept:
For a transmitting antenna of height \(h\), the line-of-sight distance is:
\[
d = \sqrt{2R_e h}
\]
and the service area on Earth is approximately circular:
\[
A = \pi d^2 = \pi (2R_e h)
\]
Step 1: Apply the area formula directly.
Given: \[ h = 81~\text{m}, \quad R_e = 6.4 \times 10^6~\text{m} \] \[ A = \pi \cdot 2R_e h = \pi \cdot 2 \cdot (6.4 \times 10^6) \cdot 81 \]
Step 2: Simplify numerical value.
\[ A = 3.1416 \times 2 \times 6.4 \times 81 \times 10^6 \] \[ 2 \times 6.4 = 12.8,\quad 12.8 \times 81 = 1036.8 \] \[ A = 3.1416 \times 1036.8 \times 10^6 \approx 3257 \times 10^6~\text{m}^2 \]
Step 3: Convert to km\(^2\).
\[ 1~\text{km}^2 = 10^6~\text{m}^2 \] \[ A \approx \frac{3257 \times 10^6}{10^6} = 3257~\text{km}^2 \] Approximating to the closest option: \[ \boxed{3251~\text{km}^2} \]
Step 1: Apply the area formula directly.
Given: \[ h = 81~\text{m}, \quad R_e = 6.4 \times 10^6~\text{m} \] \[ A = \pi \cdot 2R_e h = \pi \cdot 2 \cdot (6.4 \times 10^6) \cdot 81 \]
Step 2: Simplify numerical value.
\[ A = 3.1416 \times 2 \times 6.4 \times 81 \times 10^6 \] \[ 2 \times 6.4 = 12.8,\quad 12.8 \times 81 = 1036.8 \] \[ A = 3.1416 \times 1036.8 \times 10^6 \approx 3257 \times 10^6~\text{m}^2 \]
Step 3: Convert to km\(^2\).
\[ 1~\text{km}^2 = 10^6~\text{m}^2 \] \[ A \approx \frac{3257 \times 10^6}{10^6} = 3257~\text{km}^2 \] Approximating to the closest option: \[ \boxed{3251~\text{km}^2} \]
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