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 $tan2\alpha =\frac{30}{x}$
$\therefore$ $tan2\alpha =\frac{30}{5cot\alpha }\Rightarrow tan2\alpha =6tan\alpha$
$\Rightarrow \frac{2tan\alpha }{1-ta{n}^2\alpha }=6tan\alpha \Rightarrow tan\alpha =\sqrt{\frac{2}{3}}$


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