Multiple choice

At certain distance from a transmitting tower, a receiver tower of height $20\ m$ is used to receive direct signal. Another tower is installed beyond the first, along the same line of sight to receive the signals from the same transmitter. Its height is $44$% more than the first receiving tower. Then, the separation between the two receiving towers is (Radius of Earth = $6400\ km $)

  1. $6.4\ km$
  2. $3.2\ km$
  3. $1.6\ km$
  4. $0.8\ km$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The distance d to the horizon is sqrt(2Rh). d1 = sqrt(2 * 6400 * 0.02) = sqrt(256) = 16 km. h2 = 20 * 1.44 = 28.8 m. d2 = sqrt(2 * 6400 * 0.0288) = sqrt(368.64) = 19.2 km. Separation = d2 - d1 = 19.2 - 16 = 3.2 km.

AI explanation

The maximum line of sight distance is proportional to the square root of the receiver's height. Increasing the tower height by 44 percent makes the new height 1.44 times the original 20 m height, so the ratio of the new range to the old range is the square root of 1.44, which is 1.2. The first tower has a reception range of the square root of two times 6400 times 0.02, equaling 16 km; the second has a range of 1.2 times 16 km, equaling 19.2 km. The separation between the two towers is the difference between these ranges, 19.2 km minus 16 km, which is 3.2 km.

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