Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 6 m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30° and that of the top of the flagstaff is 60°. Find the height of the tower.
[Use $\sqrt{3}$ = 1.732] [CBSE 2011]

Answer 1

Let AB be the tower and BC be the flagstaff.

We have,
$$\begin{gathered} \mathrm{BC}=6\mathrm{m},\angle \mathrm{AOB}=30°\mathrm{and}\angle \mathrm{AOC}=60° \\ \mathrm{Let}\mathrm{AB}=h \\ \mathrm{In}∆\mathrm{AOB}, \\ \tan 30°=\frac{\mathrm{AB}}{\mathrm{OA}} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{\mathrm{OA}} \\ \Rightarrow \mathrm{OA}=h\sqrt{3}.....\left(\mathrm{i}\right) \\ \mathrm{Now},\mathrm{in}∆\mathrm{AOC}, \\ \tan 60°=\frac{\mathrm{AC}}{\mathrm{OA}} \\ \Rightarrow \sqrt{3}=\frac{\mathrm{AB}+\mathrm{BC}}{h\sqrt{3}}\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ \Rightarrow 3h=h+6 \\ \Rightarrow 3h-h=6 \\ \Rightarrow 2h=6 \\ \Rightarrow h=\frac{6}{2} \\ \Rightarrow h=3\mathrm{m} \end{gathered}$$

So, the height of the tower is 3 m.