Question

A T.V. tower at a height of 10 m is is in a region of average population density $100\pi /k{m^2}$.The number of people that can receive the transmission is nearly
A) 1,28,000
B) 64,000
C) 2,56,000
D) 32,000

Answer 1

Hint
Firstly we have to find the radius of transmission which is affected by the T.V.tower. Then we have to find the area of transmission by the T.V.tower. Finally to find the number of people that can receive the transmission ,we multiply the area with the Population density.

Complete step by step solution
According to given question:-
Height of T.V. tower ($H$) is $10m$.
We already know the radius of earth ($R$) is approximately $6400 km$.
The radius of transmission which is affected by the tower is $r$.
We know that $r = \sqrt {2RH} + \sqrt {2R{H_r}} $
Where, $H_r$ is the height of the receiver .
Now putting the value of all variables:-
$r = \sqrt {2 \times 6400 \times 10 \times {{10}^{ - 3}}} + \sqrt {2 \times 6400 \times 0} $ (here $H=0$,height of receiver)
$r = \sqrt {128} $
$r = 8\sqrt 2 $km
Now find the area of transmission.
To find area of transmission we apply A=$A = \pi {r^2}$
So, Area of transmission is equal to:-$\pi {(8\sqrt 2 )^2}$
$A = 128\pi k{m^2}$
Now we have to find total number of people who can receive the transmission
Let the total number of people be equal to $P$.
$P=area \times $population density,
$P = 128\pi \times 100\pi $,
$P = 12800{\pi ^2}$,
$P = 129,438.72$
Hence , nearly $128,000$ people can receive the transmission from the $10m$ T.V tower.
Correct answer will be option number (A).

Note
Always keep in mind to perform mathematical operations we have to check the dimension of all measurement units. It is accordance with the relation :covering radius $range = \sqrt {2HR} + \sqrt {2R{H_r}} $ where H is height of the antenna ,R is radius of the earth and ${H_r}$ is height of receiver.