Question

If the height of a tower and the distance of the point of observation from its foot, both, are increased by 10%, then the angle of elevation of its top remains ___________.

Answer 1

Let AB be the height of the tower and the initial distance of point of observation from its foot be BC.

Suppose the initial angle of elevation from the point of observation be θ.



In right ∆ABC,

$\tan \theta =\frac{\mathrm{AB}}{\mathrm{BC}}.....\left(1\right)$

If the height of a tower and the distance of the point of observation from its foot, both, are increased by 10%, then

New height of the tower, A'B' = AB + 10% of AB $=\mathrm{AB}+\frac{1}{10}\mathrm{AB}=\frac{11}{10}\mathrm{AB}$
New distance of point of observation from the foot, B'C' = BC + 10% of BC $=\mathrm{BC}+\frac{1}{10}\mathrm{BC}=\frac{11}{10}\mathrm{BC}$

Suppose the new angle of elevation from the point of observation be θ'.



In right ∆A'B'C',

$$\begin{gathered} \tan \theta \text{'}=\frac{\mathrm{A}\text{'}\mathrm{B}\text{'}}{\mathrm{B}\text{'}\mathrm{C}\text{'}} \\ \Rightarrow \tan \theta \text{'}=\frac{{\displaystyle \frac{11}{10}}\mathrm{AB}}{{\displaystyle \frac{11}{10}}\mathrm{BC}}=\frac{\mathrm{AB}}{\mathrm{BC}} \\ \Rightarrow \tan \theta \text{'}=\tan \theta \left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow \theta \text{'}=\theta \end{gathered}$$

So, the new angle of elevation of the top of the tower remains unchanged.

If the height of a tower and the distance of the point of observation from its foot, both, are increased by 10%, then the angle of elevation of its top remains __unchanged__.