The maximum distance upto which TV transmission from a TV tower of...

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 \gt \gt h$. Let$A$be a receiving station. In the limit$h\lt \lt R$, we can assume that$B A$is a tangent to the surface of the earth. Then$\angle B A O=90^{{}^{\circ}} and $so B O²=A B²+A O² \Rightarrow(R+h)^2=A B²+R² \Rightarrow \quad A B²=R²+h²+2 R h-R²=2 R h+h² as$h\lt \lt R$we neglect$h^2$and so$$\begin{aligned}& A B²=2 R h \Rightarrow A B=\sqrt{2 R h}\\ \therefore \quad & A B \propto h^{1 / 2} . \end{aligned}$$