question_answer

ABC is triangular park with \[AB=AC=100m\] A clock tower is situated at the mid-point of BC. The angles of elevation of the top of the tower at A and B are \[{{\cot }^{-1}}3.2\] and \[\cos e{{c}^{-}}^{1}\,2.6\] respectively. The height of the tower is

A) \[50\,m\]                      

B) \[25\,m\]

C) \[40\,m\]                      

D) None of these

Correct Answer: B

Solution :

DP is a clock tower standing at the middle point D of BC.

\[\angle PAD=\alpha ={{\cot }^{-1}}3.2\Rightarrow \cot \,\alpha =3.2\]

and \[\angle PBD=\beta ={{\operatorname{cosec}}^{-1}}2.6\Rightarrow \operatorname{cosec}\,\beta =2.4\]

\[\therefore \,\,\cot \beta =\sqrt{\left( \cos e{{c}^{2}}\beta -1 \right)}=\sqrt{\left( 5.76 \right)}=2.4\]

In the triangles PAD and PAD,

\[AD=h\cot \alpha =3.2h\] and \[BD=h\,\cot \beta =2.4\,h\]

In the right angled \[\Delta ABD,A{{B}^{2}}=A{{D}^{2}}+B{{D}^{2}}\]

\[\Rightarrow \,\,{{100}^{2}}=[{{(3.2)}^{2}}+{{(2.4)}^{2}}]{{h}^{2}}=16{{h}^{2}}\Rightarrow h=25m.\]