Write 'True' or 'False' and justify your answer in each of the following:
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.
Given:
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.
To do:
We have to find whether the given statement is true or false.
Solution:
Let the height of the tower$=h$ and the distance of the point of observation from its foot$=x$
Let the angle of elevation$=\alpha$
$\therefore tan\alpha=\frac{h}{x}\ \ \ \ ...........\ ( i)$
Again, if the height of the tower and the distance of the point of observation from its foot are increased by $10\ %$,
Then the new height$=H=h+\frac{h\times 10}{100}=\frac{11h}{10}$
And, the new distance of the point of observation from its foot$=X=x+\frac{x\times 10}{100}=\frac{11x}{10}$
Let the new angle of elevation $=\beta$
Then $tan\beta=\frac{H}{X}$
$=\frac{\frac{11h}{10}}{\frac{11x}{10}}$
$=\frac{h}{x}$
$=tan\alpha\ \ \ \ .........\ ( from\ i)$
$\therefore \alpha=\beta$
Thus, prove that the angle of elevation of its top remains unchanged.
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