Question

The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60°, then find the height of the flagstaff. [Use $\sqrt{3}$ = 1.732] [CBSE 2014]

Answer 1

Let BC and CD be the heights of the tower and the flagstaff, respectively.

We have,
$$\begin{gathered} \mathrm{AB}=120\mathrm{m},\angle \mathrm{BAC}=45°,\angle \mathrm{BAD}=60° \\ \mathrm{Let}\mathrm{CD}=x \\ \mathrm{In}∆\mathrm{ABC}, \\ \tan 45°=\frac{\mathrm{BC}}{\mathrm{AB}} \\ \Rightarrow 1=\frac{\mathrm{BC}}{120} \\ \Rightarrow \mathrm{BC}=120\mathrm{m} \\ \mathrm{Now},\mathrm{in}∆\mathrm{ABD}, \\ \tan 60°=\frac{\mathrm{BD}}{\mathrm{AB}} \\ \Rightarrow \sqrt{3}=\frac{\mathrm{BC}+\mathrm{CD}}{120} \\ \Rightarrow \mathrm{BC}+\mathrm{CD}=120\sqrt{3} \\ \Rightarrow 120+x=120\sqrt{3} \\ \Rightarrow x=120\sqrt{3}-120 \\ \Rightarrow x=120\left(\sqrt{3}-1\right) \\ \Rightarrow x=120\left(1.732-1\right) \\ \Rightarrow x=120\left(0.732\right) \\ \Rightarrow x=87.84\approx 87.8\mathrm{m} \end{gathered}$$

So, the height of the flagstaff is 87.8 m.