Question

A TV transmission tower antenna is at a height of $20m$. Suppose that the receiving antenna is at.

(i) Ground level

(ii) a height of $5m$

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

Answer 1

Step 1. Give Data:

Height of the TV tower $h_T=20m$

Height of receiving antenna in the first case $h_{R_1}=0m$ (Ground level)

Height of receiving antenna in the second case $h_{R_2}=5m$

Step 2. Calculation of ranges for the cases:

if $d$is the range then it is obtained by the formula,

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

Let $d_1$ and $d_2$ be the ranges for the first and second cases

Then for the first case $d_1=\sqrt{2\times 6.4\times {10}^6\times 20}+0$

$=16km$

and for the second case $d_2=\sqrt{2\times 6.4\times {10}^6\times 20}+\sqrt{2\times 6.4\times {10}^6\times 5}$

$=24km$

Step 3. Calculate the percentage change $\mathit{n}$:

It is given by $n=\frac{d_2-d_1}{d_1}\times 100$

$=\frac{24-16}{16}\times 100$

$=50\%$

Hence the increase in the range is $50\%$.