Question

TV transmission tower antenna is at a height of $20m$. How much range can it cover if the receiving antenna is a height of $25cm$?

Answer 1

Hint
By using the formula for the range covered we can calculate the value of the maximum range by substituting the value of the transmission antenna as $20m$, and the receiving antenna as $25cm$.The radius of the earth can be taken as $6400km$.
In this solution we will be using the following formula,
$\Rightarrow d = \sqrt {2R{h_T}} + \sqrt {2R{h_R}} $
where $d$ is the maximum range,
$R$ is the radius of the earth,
${h_T}$ is the height of the transmission antenna and ${h_R}$ is the height of the receiving antenna.

Complete step by step answer
The range of the TV transmission can be calculated from the height of the tower antenna, the height of the receiving antenna and the radius of the earth. It is given by the formula,
$\Rightarrow d = \sqrt {2R{h_T}} + \sqrt {2R{h_R}} $
Now in the question, we are given that the height of the tower antenna is given as, ${h_T} = 20m$. The height of the receiving antenna is given in the question as, ${h_R} = 25cm$. We can write this in meters as, ${h_R} = 25cm = 0.25m$. The radius of the earth can be taken to be approximately $R = 6400km$, which in meter is, $R = 6.4 \times {10^6}m$
Therefore substituting these formulas, we will get the value of the maximum range as,
$\Rightarrow d = \sqrt {2 \times 6.4 \times {{10}^6} \times 20} + \sqrt {2 \times 6.4 \times {{10}^6} \times 0.25} $.
Now calculating under the roots we get,
$\Rightarrow d = \sqrt {2.56 \times {{10}^8}} + \sqrt {3.2 \times {{10}^6}} $
The values on removing the root are,
$\Rightarrow d = 16000 + 1788.854$
On adding we get the range as,
$\Rightarrow d = 17788.854m$
So this value in kilometer is approximately, $d = 17.78km$
Therefore, the range covered by the TV antenna is $17.78km$.

Note
So the radius up to which the TV transmission tower antenna range is covered is directly proportional to the height of the antenna of both the transmission tower and the receiving antenna. Using this range as the radius we can also calculate the coverage area of the antenna.