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

To do: To prove that the angle of elevation of its top remains unchanged.

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.
 

Updated on: 10-Oct-2022

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