Question:
The heights of the transmitting and receiving antennas are 33.8 m and 64.8 m respectively. The maximum distance between the antennas for satisfactory communication in line of sight mode is:
(Radius of the earth = 6400 km)
The heights of the transmitting and receiving antennas are 33.8 m and 64.8 m respectively. The maximum distance between the antennas for satisfactory communication in line of sight mode is:
(Radius of the earth = 6400 km)
(Radius of the earth = 6400 km)
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When calculating the line of sight distance between antennas, remember to account for the heights of both antennas and use the Earth's radius in the formula.
Updated On: Mar 11, 2025
- 20.8 km
- 28.8 km
- 49.6 km
- 57.6 km
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The Correct Option is C
Solution and Explanation
To determine the maximum distance between the transmitting and receiving antennas for satisfactory communication in line of sight mode, we can use the formula for the line of sight distance between two antennas:
\[
d = \sqrt{2R h_t} + \sqrt{2R h_r}
\]
where:
\( d \) is the maximum distance between the antennas,
\( R \) is the radius of the Earth,
\( h_t \) is the height of the transmitting antenna,
\( h_r \) is the height of the receiving antenna.
Given Data:
Height of transmitting antenna, \( h_t = 33.8 \, \text{m} \)
Height of receiving antenna, \( h_r = 64.8 \, \text{m} \)
Radius of the Earth, \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)
Calculation: 1. Calculate the distance contributed by the transmitting antenna: \[ d_t = \sqrt{2R h_t} = \sqrt{2 \times 6400 \times 10^3 \times 33.8} \] \[ d_t = \sqrt{2 \times 6400 \times 33.8 \times 10^3} = \sqrt{432640 \times 10^3} \approx 657.8 \times 10^{1.5} \approx 657.8 \times 31.62 \approx 20800 \, \text{m} = 20.8 \, \text{km} \] 2. Calculate the distance contributed by the receiving antenna: \[ d_r = \sqrt{2R h_r} = \sqrt{2 \times 6400 \times 10^3 \times 64.8} \] \[ d_r = \sqrt{2 \times 6400 \times 64.8 \times 10^3} = \sqrt{829440 \times 10^3} \approx 910.8 \times 10^{1.5} \approx 910.8 \times 31.62 \approx 28800 \, \text{m} = 28.8 \, \text{km} \] 3. Sum the distances to get the maximum line of sight distance: \[ d = d_t + d_r = 20.8 \, \text{km} + 28.8 \, \text{km} = 49.6 \, \text{km} \] Final Answer:
The maximum distance between the antennas for satisfactory communication is: \[ \boxed{49.6 \, \text{km}} \] This corresponds to option (3).
\( d \) is the maximum distance between the antennas,
\( R \) is the radius of the Earth,
\( h_t \) is the height of the transmitting antenna,
\( h_r \) is the height of the receiving antenna.
Given Data:
Height of transmitting antenna, \( h_t = 33.8 \, \text{m} \)
Height of receiving antenna, \( h_r = 64.8 \, \text{m} \)
Radius of the Earth, \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} \)
Calculation: 1. Calculate the distance contributed by the transmitting antenna: \[ d_t = \sqrt{2R h_t} = \sqrt{2 \times 6400 \times 10^3 \times 33.8} \] \[ d_t = \sqrt{2 \times 6400 \times 33.8 \times 10^3} = \sqrt{432640 \times 10^3} \approx 657.8 \times 10^{1.5} \approx 657.8 \times 31.62 \approx 20800 \, \text{m} = 20.8 \, \text{km} \] 2. Calculate the distance contributed by the receiving antenna: \[ d_r = \sqrt{2R h_r} = \sqrt{2 \times 6400 \times 10^3 \times 64.8} \] \[ d_r = \sqrt{2 \times 6400 \times 64.8 \times 10^3} = \sqrt{829440 \times 10^3} \approx 910.8 \times 10^{1.5} \approx 910.8 \times 31.62 \approx 28800 \, \text{m} = 28.8 \, \text{km} \] 3. Sum the distances to get the maximum line of sight distance: \[ d = d_t + d_r = 20.8 \, \text{km} + 28.8 \, \text{km} = 49.6 \, \text{km} \] Final Answer:
The maximum distance between the antennas for satisfactory communication is: \[ \boxed{49.6 \, \text{km}} \] This corresponds to option (3).
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