Question

The maximum distance up to which TV transmission from a TV tower of height h can be received is proportional to

Answer 1

Step 1: Given data

The height of the TV tower is $h$.

Step 2: Solution

We need to calculate the maximum distance the signals travel from the tower considering the fact that the surface of the earth is curved.

Step 3: Construct the conditions using the fact the earth is curved

$AO$ is the radius of the earth $R$

$AB$ is the maximum distance the signal can travel, let it be $D$

We can clearly see $OB\hspace{0.17em}=h+R$

And we can observe $\angle OAB=90°$

Step 4: Using Pythagoras's theorem

From Pythagoras theorem $O{B}^2=O{A}^2+A{B}^2$

⇒ ${(h+R)}^2=d^2+R^2$

⇒ $d^2=h^2+R^2+2Rh-R^2$

⇒ $d^2=h^2+2Rh$

As $R>>h$ we can neglect $h^2$

⇒ $d^2=2hR$

⇒ $d=\sqrt{2hR}$

⇒ The maximum distance the signals travel is $d=\sqrt{2hR}$

Now, we can easily see that $d$∝ $\sqrt{h}$.

Thus the maximum distance up to which TV transmission from a TV tower of height h can be received is proportional to the square root of h.