Trigonometric Equations Question 178
Question: 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]
Options:
A) 50 metres
B) $ 50,\sqrt{3} $ metres
C) $ 50,\sqrt{2} $ metres
D) None of these
Show Answer
Answer:
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,( \frac{\sin \alpha }{\sin 3\alpha \sin 2\alpha } ) $ and $ 200=h,( \frac{\sin \alpha }{\sin 2\alpha \sin \alpha } ) $ 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}}( \frac{\pi }{6} )\Rightarrow \alpha =\frac{\pi }{6} $ or $ \alpha =\frac{5\pi }{6} $ Hence, $ h=200\sin \frac{\pi }{3}=200\frac{\sqrt{3}}{2}=100\sqrt{3} $ , {from (ii)} .
BETA
