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. |
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