The angles of elevation of the top of a tower from two points A and B lying on the horizontal through the foot of the tower are respectively 15° and 30°. If A and B are on the same side of the tower and AB = 48 metre, then the height of the tower is :

Tower = PQ = h metre

QB = x metre

From $riangle mathrm{APQ}, an 15^{circ}=frac{h}{x+48}$

$2-sqrt{3}=frac{h}{x+48}$...(i)

From $Delta mathrm{PQB}, an 30^{circ}=frac{h}{x}$

$Rightarrow frac{1}{sqrt{3}}=frac{h}{x}$

$Rightarrow sqrt{3} h=x$

$ldots$ (ii)

$Rightarrow 2-sqrt{3}=frac{h}{sqrt{3} h+48}$

$Rightarrow 2 sqrt{3} h-3 h+(2-sqrt{3}) 48=h$

$Rightarrow h+3 h-2 sqrt{3} h=(2-sqrt{3}) imes 48$

$Rightarrow 2 h(2-sqrt{3})=48 imes(2-sqrt{3})$

$Rightarrow h=frac{48}{2}=24$ metre