Question

A flag staff on the top of the tower of $80m$ height subtends an angle ${\tan }^{-1}\frac{1}{9}$ at a point on the ground $100m$ from the foot of the tower. The height of the flag staff is

Answer 1

$\begin{array}{c}Wehave\\ {\tan }^{-1}\left(\frac{1}{9}\right)=\alpha \\ \tan \alpha =\frac{1}{9}\\ In\Delta BCD\\ \tan \beta =\frac{80}{100}\\ In\Delta ACD\\ \tan \left(\alpha +\beta \right)=\frac{80+x}{100}\\ \frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }=\frac{80+x}{100}\\ \frac{\frac{1}{9}+\frac{80}{100}}{1-\frac{80}{900}}=\frac{80+x}{100}\\ \frac{\frac{820}{900}}{\frac{720}{900}}=\frac{80+x}{100}\\ 1=\frac{80+x}{100}\\ 80+x=100\\ \therefore x=100-80\\ =20\\ Hence,theheightoftheflagstaffis20.\end{array}$