If the height of the antenna is $h$, then the maximum distance upto which the TV transmission for a TV tower can be received is proportional to $h^{1 / 2}$.


Suppose that the height of the TV antena be $h$ and the radius of the earth be $R$ and $R>>h$. Let $A$ be a receiving station. In
the limit $h< < R$, we can assume that $B A$ is a tangent to the surface of the earth. Then $\angle B A O=90^{\circ}$ and so
$$
B O^2=A B^2+A O^2 \Rightarrow(R+h)^2=A B^2+R^2
$$

$$
\Rightarrow \quad A B^2=R^2+h^2+2 R h-R^2=2 R h+h^2
$$
as $h< < R$ we neglect $h^2$ and so
$$
\begin{aligned}
& A B^2=2 R h \Rightarrow A B=\sqrt{2 R h} \\
\therefore \quad & A B \propto h^{1 / 2} .
\end{aligned}
$$