Question

A flag - staff of $5$ meters high stands on a building of $25$ meters height. For an observer at a height of $30$ meters, the flag-staff and the building subtend equal angles. The distance of the observer from the top of the flag - staff is

• A. $\frac{5 \sqrt{3}}{2}\text{ m}$
• B. $5\sqrt{\frac{3}{2}}\text{ m}$
• C. $5\sqrt{\frac{2}{3}}\text{ m}$
• D. None of these

Answer 1

Answer: (B)

We have $tan \alpha = \frac{5}{x}$ and $tan 2 \alpha = \frac{30}{x}$
$\therefore $ $tan 2 \alpha =\frac{30}{5 cot ⁡ \alpha } \, \Rightarrow \, tan ⁡ 2 \alpha =6tan ⁡ \alpha $
$\Rightarrow \frac{2 tan \alpha }{1 - tan^{2} \alpha }=6tan\alpha \Rightarrow tan\alpha =\sqrt{\frac{2}{3}}$


Hence, $x=\frac{5}{tan \alpha }\Rightarrow x=5\sqrt{\frac{3}{2}}m$