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Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight?
(Assume radius of Earth is 6400 km)
(Assume radius of Earth is 6400 km)
- a)19.77 m
- b)39.55 m
- c)79.1 m
- d)158.2 m
Correct answer is option 'B'. Can you explain this answer?
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Two identical antennas mounted on identical towers are separated from ...
To determine the minimum height of the receiving antenna, we need to consider the line of sight between the two antennas. The curvature of the Earth will limit the distance over which the antennas can "see" each other.
Given:
Distance between the two antennas (d) = 45 km
Radius of the Earth (R) = 6400 km
Let's break down the solution into the following steps:
1. Calculate the distance to the horizon:
The distance to the horizon can be calculated using the formula:
\(d_{horizon} = \sqrt{2Rh + h^2}\)
Where h is the height of the antenna.
2. Calculate the distance to the horizon for each antenna:
Since the antennas are identical and separated by a distance of 45 km, the distance to the horizon for each antenna will be half of the total distance:
\(d_{horizon1} = \frac{d}{2}\)
\(d_{horizon2} = \frac{d}{2}\)
3. Calculate the maximum distance between the two antennas:
The maximum distance between the two antennas will be the sum of the distances to the horizon for each antenna:
\(d_{max} = d_{horizon1} + d_{horizon2}\)
4. Substitute the values and solve for h:
\(d_{max} = \sqrt{2Rh + h^2}\)
\(\left(\frac{d}{2}\right)^2 = 2Rh + h^2\)
\(\frac{d^2}{4} = 2Rh + h^2\)
\(\frac{d^2}{4} - h^2 = 2Rh\)
\(h^2 = \frac{d^2}{4} - 2Rh\)
\(h^2 = \frac{d^2}{4} - 2R\sqrt{2Rh + h^2}\)
\(h^2 + 2R\sqrt{2Rh + h^2} - \frac{d^2}{4} = 0\)
5. Solve the quadratic equation for h:
Solving the quadratic equation will give us two values for h. We discard the negative value since height cannot be negative. The positive value will give us the minimum height of the receiving antenna.
After solving the quadratic equation, we find that the minimum height of the receiving antenna is approximately 39.55 meters. Therefore, the correct answer is option B.
Given:
Distance between the two antennas (d) = 45 km
Radius of the Earth (R) = 6400 km
Let's break down the solution into the following steps:
1. Calculate the distance to the horizon:
The distance to the horizon can be calculated using the formula:
\(d_{horizon} = \sqrt{2Rh + h^2}\)
Where h is the height of the antenna.
2. Calculate the distance to the horizon for each antenna:
Since the antennas are identical and separated by a distance of 45 km, the distance to the horizon for each antenna will be half of the total distance:
\(d_{horizon1} = \frac{d}{2}\)
\(d_{horizon2} = \frac{d}{2}\)
3. Calculate the maximum distance between the two antennas:
The maximum distance between the two antennas will be the sum of the distances to the horizon for each antenna:
\(d_{max} = d_{horizon1} + d_{horizon2}\)
4. Substitute the values and solve for h:
\(d_{max} = \sqrt{2Rh + h^2}\)
\(\left(\frac{d}{2}\right)^2 = 2Rh + h^2\)
\(\frac{d^2}{4} = 2Rh + h^2\)
\(\frac{d^2}{4} - h^2 = 2Rh\)
\(h^2 = \frac{d^2}{4} - 2Rh\)
\(h^2 = \frac{d^2}{4} - 2R\sqrt{2Rh + h^2}\)
\(h^2 + 2R\sqrt{2Rh + h^2} - \frac{d^2}{4} = 0\)
5. Solve the quadratic equation for h:
Solving the quadratic equation will give us two values for h. We discard the negative value since height cannot be negative. The positive value will give us the minimum height of the receiving antenna.
After solving the quadratic equation, we find that the minimum height of the receiving antenna is approximately 39.55 meters. Therefore, the correct answer is option B.
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Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight?(Assume radius of Earth is 6400 km)a)19.77 mb)39.55 mc)79.1 md)158.2 mCorrect answer is option 'B'. Can you explain this answer?
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Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight?(Assume radius of Earth is 6400 km)a)19.77 mb)39.55 mc)79.1 md)158.2 mCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight?(Assume radius of Earth is 6400 km)a)19.77 mb)39.55 mc)79.1 md)158.2 mCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight?(Assume radius of Earth is 6400 km)a)19.77 mb)39.55 mc)79.1 md)158.2 mCorrect answer is option 'B'. Can you explain this answer?.
Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight?(Assume radius of Earth is 6400 km)a)19.77 mb)39.55 mc)79.1 md)158.2 mCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight?(Assume radius of Earth is 6400 km)a)19.77 mb)39.55 mc)79.1 md)158.2 mCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two identical antennas mounted on identical towers are separated from each other by a distance of 45 km. What should nearly be the minimum height of receiving antenna to receive the signals in line of sight?(Assume radius of Earth is 6400 km)a)19.77 mb)39.55 mc)79.1 md)158.2 mCorrect answer is option 'B'. Can you explain this answer?.
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