Question

A TV transmission tower antenna is at a height of $20\text{m}$. Suppose that the receiving antenna is at -
$\left(i\right)$ Ground Level
$\left(ii\right)$ a height of $5\text{m}$

The increase in antenna range in case $\left(ii\right)$ relative to case $\left(i\right)$ is $n\%$. The value of $n$, to the nearest integer, is ______.

Answer 1

For calculation of range from a tower,

$d=\sqrt{2R{h}_T}+\sqrt{2R{h}_R}$

Where,
$h_T=$ height of tower
$h_R=$ height of receiver
$R=$ radius of Earth.

For $\left(i\right)$,

$h_T=20\text{m},h_R=0$

$\therefore d_i=\sqrt{2\times 6400\times {10}^3\times 20}=16\text{km}$

For $\left(ii\right)$,

$h_T=20\text{m},h_R=5\text{m}$

$\because d_{ii}=\sqrt{2\times 6400\times {10}^3\times 20}+\sqrt{2\times 6400\times {10}^3\times 5}$

= 16 + 8 = 24 km

$\therefore \%$ change in range,

$=\frac{d_{ii}-d_i}{d_i}\times 100$

$=\frac{24-16}{16}\times 100=50\%=n\%$

So, $n=50$