A spherical balloon of radius r subtends an angle of 60° at the eye of an observer. If the angle of elevation of its…
CBSE Class 10 Maths PYQ · Applications of Trig · Double Triangle · 5 Marks · March 2023 · Standard
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355 Marks · March 2023 · Standard
A spherical balloon of radius $r$ subtends an angle of $60^\circ$ at the eye of an observer. If the angle of elevation of its centre is $45^\circ$ from the same point, then prove that height of the centre of the balloon is $\sqrt{2}$ times its radius.
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1 mark for correct figure Let Point B represents observer. $\therefore \angle QBP = 60^\circ$; $\angle ABO = 45^\circ$ Using geometry $\angle PBO = \frac{1}{2} \times 60^\circ = 30^\circ$ Now, $\frac{r}{OB} = \sin 30^\circ = \frac{1}{2} \Rightarrow OB = 2r$ (i) Also $\frac{OA}{OB} = \sin 45^\circ = \frac{1}{\sqrt{2}} \Rightarrow OB = OA \sqrt{2}$ (ii) Using (i) and (ii) $OA = \sqrt{2} r$ or height of center of balloon = $\sqrt{2} r$ units
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