question_answer

A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 metres and the distance between B and C is 100 metres, then the height of balloon is given by [Roorkee 1989]

A) 50 metres

B) \[50\,\sqrt{3}\] metres

C) \[50\,\sqrt{2}\] metres

D) None of these

Correct Answer: D

Solution :

\[x=h\cot 3\alpha \]    .....(i) \[(x+100)=h\cot 2\alpha \] ......(ii) \[(x+300)=h\cot \alpha \] ......(iii) From (i) and (ii), \[-100=h\,(\cot 3\alpha -\cot 2\alpha ),\] From (ii) and (iii), \[-200=h(\cot 2\alpha -\cot \alpha ),\] \[100=h\,\left( \frac{\sin \alpha }{\sin 3\alpha \sin 2\alpha } \right)\] and \[200=h\,\left( \frac{\sin \alpha }{\sin 2\alpha \sin \alpha } \right)\] or \[\frac{\sin 3\alpha }{\sin \alpha }=\frac{200}{100}\Rightarrow \frac{\sin 3\alpha }{\sin \alpha }=2\] \[\Rightarrow \] \[3\sin \alpha -4{{\sin }^{3}}\alpha -2\sin \alpha =0\] \[\Rightarrow \] \[4{{\sin }^{3}}\alpha -\sin \alpha =0\Rightarrow \sin \alpha =0\] or \[{{\sin }^{2}}\alpha =\frac{1}{4}={{\sin }^{2}}\left( \frac{\pi }{6} \right)\Rightarrow \alpha =\frac{\pi }{6}\] Hence, \[h=200\sin \frac{\pi }{3}=200\frac{\sqrt{3}}{2}=100\sqrt{3}\],  {from (ii)} .