Question

A flagstaff on the top of a tower $80$ m high subtends an angle of ${\tan }^{-1}\left(\frac{1}{9}\right)$ at a point on the ground $100$ m from the foot of a tower The height of the flagstaff is

• A. $20m$
• B. $30m$
• C. $40m$
• D. $25m$

Answer 1

The correct option is A $20m$

From the figure,

$\tan \theta =\frac{AB}{OA}=\frac{80}{100}=\frac{4}{5}$

$\tan \theta =\frac{1}{9}$

$\tan (\theta +\varphi )=\frac{80+x}{100}$

$\Rightarrow \frac{\tan \theta +\tan \varphi }{1-\tan \theta \tan \varphi }=\frac{80+x}{100}=\frac{\frac{4}{5}+\frac{1}{9}}{1-\frac{4}{5}\times \frac{1}{9}}=\frac{80+x}{100}$

$\therefore x=20m$