Question

A TV transmission tower of antenna is at a height of 20 m. How much service area can it cover if the receiving antenna is (i) at ground level (ii) at a height of 25 m ? Calculate the percentage increase in area covered in case (ii) relative to case (i).

Answer 1

Given, height of antenna h = 20 m
Radius of earth = `6.4 xx 10^(6)m`
At the ground level,
(i) Range = `sqrt(2hR)=sqrt(2 xx 20 xx 6.4 xx 10^(6))`
= 16000m = 16 km
Area covered A = `pi("range")^(2)`
`=3.14 xx 16 xx 16 = 803. 84 km^(2)`
(iii) At a height of H = 25 m from ground level
Range = `sqrt(2hR)+ sqrt(2HR)`
`=sqrt(2xx20 xx6.4 xx 10^(6))+sqrt(2xx25xx6.4 xx10^(6))`
`=16 xx 10^(3) +17.9 xx 10^(3)`
`=33.9 xx 10^(3)m`
`=33.9 km`
Area covered = `pi ("Range")^(2)`
`3.14 xx 33.9 xx 33.9`
`= 3608.52 km^(2)`
Percentage increase in area `=("Difference in area")/("Initial area")xx100`
`=((3608.52 - 803.84))/(803.84)xx100`
`=348.9 %`
Thus, the percentage increase in area covered is `348.9%`