question_answer

The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 100 m from its base is \[45{}^\circ .\]If the angle of elevation of the top of complete pillar at the same point is to be \[60{}^\circ ,\] then the height of the incomplete pillar is to be increased by

A)  \[50\sqrt{2}\,\,\text{m}\]

B)  100 m

C)  \[100\,\,(\sqrt{3}-1)\,\,\text{m}\]

D) \[100\,\,(\sqrt{3}+1)\,\,\text{m}\]

Correct Answer: C

Solution :

Let the height of the incomplete piller be x m and the increasing height be PC = h m.

In \[\Delta ABC,\]

\[\tan 45{}^\circ =\frac{x}{100}\]

\[\Rightarrow \]   \[x=100\,\,\text{m}\]

and in \[\Delta APB,\]

\[\tan 60{}^\circ =\frac{x+h}{100}\]

\[\Rightarrow \]   \[x+h=100\sqrt{3}\]

\[\Rightarrow \]   \[h=100\sqrt{3}-x=100\sqrt{3}-100\]

\[\Rightarrow \]   \[h=100\,\,(\sqrt{3}-1)\,\,\text{m}\]