Write 'True' or 'False' and justify your answer in each of the following:
The angle of elevation of the top of a tower is $ 30^{\circ} $. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.
Given:
The angle of elevation of the top of a tower is \( 30^{\circ} \). If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.
To do:
We have to find whether the given statement is true or false.
Solution:
The given angle of elevation $=30^o$.
Let the height of the tower$=h$, and the viewer be at a distance of $x$ from the foot of the tower.
Then,
$\frac{h}{x}=tan30^o=\frac{1}{3}\ ........\ ( i)$
If the height of the tower is doubled then the new height $=2h$.
Let the angle of elevation of the top be $\theta$.
Then, $tan\theta =\frac{2h}{x}=2\times \frac{1}{3}=\frac{2}{3}\ ........( ii)$
But if the angle of elevation doubles then it should be $=\theta =2\times 30^o=60^o$.
Then, $tan\theta =tan60^o=3\ ........\ ( iii)$.
Comparing $( ii)$ & $( iii)$, there is a contradiction.
Thus, we came to know, if the height of the tower is doubled, then the angle of elevation of its top will not be doubled.
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