Question

If the sum of the heights of transmitting and receiving antennas in the line of sight of communication is fixed at \( 160 \, m \), then the maximum range of LOS communication is \(\dots\dots\dots\) km.
(Take radius of Earth \( = 6400 \, km \))

Hint:

Using symmetry (making transmitting and receiving antennas of equal height) always yields the maximum range for a constrained total antenna material/height.

Answer 1

Correct Answer: 64

Step 1: Understanding the Concept:
The maximum line-of-sight (LOS) distance \( d \) between two antennas of heights \( h_t \) and \( h_r \) is given by the sum of their individual horizons. To maximize this distance for a fixed sum of heights, the heights should be equal.
Step 2: Key Formula or Approach:
1. LOS Distance: \( d = \sqrt{2Rh_t} + \sqrt{2Rh_r} \).
2. Constraint: \( h_t + h_r = 160 \, m \).
Step 3: Detailed Explanation:
1. Maximization condition:
The function \( f(h_t, h_r) = \sqrt{h_t} + \sqrt{h_r} \) is maximized when \( h_t = h_r \) for a constant sum \( h_t + h_r \).
Thus, \( h_t = h_r = \frac{160}{2} = 80 \, m \).
2. Calculate Range d:
\[ d = \sqrt{2 \times 6400 \times 10^3 \times 80} + \sqrt{2 \times 6400 \times 10^3 \times 80} \]
\[ d = 2 \times \sqrt{2 \times 6.4 \times 10^6 \times 80} \]
\[ d = 2 \times \sqrt{1024 \times 10^6} \]
\[ d = 2 \times 32 \times 10^3 = 64000 \, m \]
3. Convert to km:
\[ d = 64 \, km \]
Step 4: Final Answer:
The maximum range is 64 km.