question_answer

The angles of elevation of the top of a tower from two points which are at distances of 10 m and 5 m from the base of the tower and in the same straight line with it are complementary. The height of the tower is

A) \[5\,\,m\]                                   

B) \[15\,\,m\]

C) \[\sqrt{50}\,\,m\]

D) \[\sqrt{75}\,\,m\]

Correct Answer: C

Solution :

Given that, angles are complementary.

Let h be the height of the tower.

Now, in \[\Delta PBC,\]

\[\tan \theta =\frac{h}{5}\]                     (i)

and in \[\Delta PAC,\]

\[\tan \,\,(90{}^\circ -\theta )=\frac{h}{10}\]

\[\Rightarrow \]   \[\cot \theta =\frac{h}{10}\]                    (ii)

On multiplying Eqs. (i) and (ii), we get

\[\tan \theta \cdot \cot \theta =\frac{h}{5}\times \frac{h}{10}\]

\[\Rightarrow \]   \[\frac{{{h}^{2}}}{50}=1\]\[\Rightarrow \]\[h=\sqrt{50}\,\,m\]

which is the required height of the tower.