Question

Two boats approach the light house in mid-sea from opposite directions. The angles of elevation of the top of the light house from two boats are ${30}^{\circ }$ and ${45}^{\circ }$. If the distance between the two boats is 100 m find the height of the light house.

Answer 1

Let $AB=x,thenAC=100-x,OA=h$

In $△OAB$,

$\tan {45}^{\circ }=\frac{OA}{AB}=\frac{h}{x}$

$\Rightarrow 1=\frac{h}{x}$

$\Rightarrow h=x$-----------------(1)

In $△OAC$,

$\tan {30}^{\circ }=\frac{1}{\sqrt{3}}=\frac{OA}{AC}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{100-x}$

$\Rightarrow 100-x=\sqrt{3}h$--------------(2)

Eliminating 'x' from Equation(2) using (1), we get

$100-h=\sqrt{3}h$

$\Rightarrow h=\frac{100}{(\sqrt{3}+1)}=50(\sqrt{3}-1)m$