Question

The angle of elevation of a cloud from a point $60m$ above a lake is $30°$ and the angle of depression of the reflection of the cloud in the lake is $60°$ find the height of the cloud from the surface of the lake?

Answer 1

Step 1- Finding the distance between point $OandD$

Consider a right-angled triangle $△AOD$

from the figure,

$OB=CD=60m$

$A\text{'}B=AB=\left(60+H\right)m$

In$△AOD$

$$\begin{gathered} \Rightarrow \tan \left(30°\right)=\frac{AO}{OD} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{H}{OD} \\ \Rightarrow OD=\sqrt{3}Hm \end{gathered}$$

Step 2- Finding the height of the cloud above the lake

,$$\begin{gathered} In△A\text{'}OD \\ \Rightarrow \tan \left(60°\right)=\frac{A\text{'}O}{OD} \\ \Rightarrow \tan \left(60°\right)=\frac{A\text{'}B+OB}{OD} \\ \Rightarrow \sqrt{3}=\frac{60+H+60}{\sqrt{3}H} \\ \Rightarrow 3H=120+H \\ \Rightarrow 3H-2H=120 \\ \Rightarrow 2H=120 \\ \Rightarrow H=\frac{120}{2} \\ \Rightarrow H=60m \end{gathered}$$

Hence, the height of the cloud above the lake $(OB+H)=\left(60+H\right)m$

$$\begin{gathered} =60+60 \\ =120m \end{gathered}$$