question_answer

The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is \[30{}^\circ \] and the angle of depression of its shadow in water of lake is \[60{}^\circ \]. Find the height of the cloud from the surface of water.

Answer:

In \[\Delta \text{ }CMP\]

\[\tan \,\,30{}^\circ =\frac{CM}{PM}\]

\[\frac{1}{\sqrt{3}}=\frac{h}{PM}\] or \[PM=\sqrt{3}h\]                                            ?(i)

In \[\Delta \,PMC'\]

\[\tan \,\,60{}^\circ =\frac{C'M}{PM}\]

\[=\frac{h+60+60}{PM}=\sqrt{3}\]

Or                           \[PM=\frac{h+120}{\sqrt{3}}\]                                            ?(ii)

From (i) and (ii)

\[\sqrt{3}h=\frac{h+120}{\sqrt{3}}\]

\[\Rightarrow \]                           \[3h=h+120\]

\[2h=120\Rightarrow h=60\,\,m\]

Height of cloud from surface of water \[=h+60=60+60=120\text{ }m\].