question_answer

Two houses are collinear with the base of a tower and are at distance 3 m and 12 m from the base of the tower. The angles of elevation from these two houses of the top of the tower are complementary. What is the height of the tower?

A) 4 m      

B)  6 m 

C)  7.5 m

D)  36 m

Correct Answer: B

Solution :

  

Let h be the height of the tower and \[\angle CBD=\theta \]

\[\angle DAC=90{}^\circ -\theta \]

\[\therefore \] In \[\Delta BCD,\]

\[\tan \theta =\frac{CD}{BC}\]

\[\Rightarrow \]   \[\tan \theta =\frac{h}{3}\]                                 ?(i)

and in \[\Delta ACD,\]

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

\[\Rightarrow \]\[\cot \theta =\frac{h}{12}\]                       ?(ii)

From Eqs. (i) and (ii), we get

\[\tan \theta \cdot \cot \theta =\frac{h}{3}\cdot \frac{h}{12}\]

\[\Rightarrow \]   \[{{h}^{2}}=36\]\[\Rightarrow \]\[h=6\,\,\text{m}\]