If the sum of the heights of transmitting and receiving antennas in line of sight of communication is fixed at h, show that the range is maximum when the two antennas have a height `h//2` each.

Explanation

The range (r) of line of sight communication
between two antennas (during space wave
propagation) is given by
`r = sqrt(2Rh_(T)) + sqrt(2Rh_(R))`
where `h_(T) and h_(R)` are the height of transmitting
and receiving antennas.
Here, `h_(T) + h_(R) = h. If h_(T) = H, then h_(R) = h-H`
Then `r = sqrt(2R)[sqrt(H) + sqrt(h-H)]`
For r to be maximum, `(dr)/(dH) = 0`
`:. (dr)/(dH) = 0 = sqrt(2R) [ 1/(2sqrtH) + (1)/(2sqrt (h-H)xx(-1))]`
or `1/(2sqrtH) - 1/(2sqrt(h-H)) = 0`
or `2sqrt(H) = 2 sqrt(h-H)`
or `H = h-H or h = 2H`
or `H = h//2`
Hence, for maximum range,
`h_(T) = h_(R) = h//2`.