Question

A T.V. antenna is of height $h$ metre. Show that it can be used to transmit the signal up to the distance $d=\sqrt{2hR}$ on the earth surface, where $R$ is the radius of earth.

Answer 1

Let $O$ be the center and $R$ is the radius of the earth. $PQ$ is a T.V. antenna of height $h$. Two tangents $QT$ and $QS$ are drawn from the top $Q$ of tower. T.V. signal can reach to the point $S$ and $T$ only not at other area:

$\angle QTO=90[TO\perp QT]$

In $\Delta QTO$,

${OQ}^2={OT}^2+{TQ}^2$

$OQ=R+h$, $QT=d$, $OT=R$,

${(R+h)}^2=R^2+d^2$

$R^2+h^2+2Rh=R^2+d^2$

$d^2=h^2+2Rh$ $h⋘R$ neglect

$d^2=2Rh$

$d=\sqrt{2Rh}$