Question

The angle of elevation of a cloud from a point $P$, $60m$ above the surface of a lake is ${30}^0$ and the angle of depression of the reflection of the cloud in the lake at $P$ is ${60}^0$. Find the height of the cloud from the surface of the lake.

Answer 1

Let AO= H

CD= OB = 60 m

A'B=AB=(60+H)m

In $\Delta AOB,tan{30}^{\circ }=\frac{OA}{OD}=\frac{H}{OD}$

$H=\frac{OD}{\sqrt{3}},OD=\sqrt{3}H$

$\Delta AOD,tan{60}^{\circ }=\frac{OA}{OD}=\frac{(OB+BA)}{OD}$

$\sqrt{3}=\frac{(60+60+H)}{\sqrt{3}H}\Rightarrow \frac{120H}{\sqrt{3}H}$

$\Rightarrow 1200+H=3H\Rightarrow 2H=120\Rightarrow H=60m$

Height of the cloud above the lake = AB+A'B= 60+60 = 120 m