question_answer

A spherical balloon of radius r subtends angle 60° at the eye of an observer. If the angle of elevation of its centre is 60° and h is the height of the centre of the balloon, then which one of the following is correct?

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\[\Delta AOC\], \[\Rightarrow \,\,\,\,\,\,\sin \left( \frac{{{60}^{{}^\circ }}}{2} \right)=\frac{OC}{AO}\]                              ? (ii) From Eqs. (i) and (ii), \[\frac{OB}{\sin {{60}^{{}^\circ }}}=\frac{OC}{\sin {{30}^{{}^\circ }}}\] \[\Rightarrow \,\,\,\,\,\frac{h}{\sqrt{3}}=\frac{r}{1}\Rightarrow \]