question_answer

A spherical balloon of radius r subtends angle \[60{}^\circ \]at the eye of an observer. If the angle of elevation of its centre is \[60{}^\circ \]and h is the height of the centre of the balloon, then which one of the following is correct?  [CDS 2013]

A) \[h=r\]              

B) \[h=\sqrt{2}r\]

C) \[h=\sqrt{3}r\]

D) \[h=2r\]

Correct Answer: C

Solution :

In \[\Delta ABO,\]\[\sin 60{}^\circ =\frac{OB}{AO}\]

\[\Rightarrow \]   \[AO=\frac{OB}{\sin 60{}^\circ }\]                                (i)

Now, in \[\Delta AOC,\]

\[\sin \frac{60{}^\circ }{2}=\frac{OC}{AO}\]

\[\Rightarrow \]   \[AO=\frac{OC}{\sin 30{}^\circ }\]                    (ii)

From Eqs. (i) and (ii), we get

\[\frac{OB}{\sin 60{}^\circ }=\frac{OC}{\sin 30{}^\circ }\]\[\Rightarrow \]\[\frac{h}{\frac{\sqrt{3}}{2}}=\frac{r}{\frac{1}{2}}\]

\[\therefore \]      \[h=\sqrt{3}r\]