Question

Establish relation $d=\sqrt{2hR}$ between height h of the TV antenna and distance d upto which TV signal can be transmitted, where R = Radius of earth.

Answer 1

Let O be the centre and R is the radius of the earth PQ is a T.V. antenna 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 if
$\angle QTO=90[TO\perp QT]$
In $\Delta QTO$,
$O{Q}^2=O{T}^2+T{Q}^2$
$OQ=R+h,QT=d,OT=R$,
$(R+h{)}^2=R^2+d^2$
$R^2+h^2+2Rh=R^2+d^2$
$d^2=h^2+2RH$ h <<< R neglect
$d^2=2Rh$
$d^2=\sqrt{2Rh}$