question_answer

\[\angle CAB=\alpha =\]

A) \[{{\cos }^{-1}}\left( \frac{1}{5} \right)\]

B) \[{{\cos }^{-1}}\left( \frac{2}{5} \right)\]

C) \[{{\cos }^{-1}}\left( \frac{\sqrt{3}}{2} \right)\]

D) \[{{\cos }^{-1}}\left( \frac{4}{5} \right)\]

Correct Answer: C

Solution :

\[A{{B}^{2}}={{\left( AD \right)}^{2}}+{{\left( BD \right)}^{2}}={{\left( 30\sqrt{3} \right)}^{2}}+{{\left( 30 \right)}^{2}}\] \[=2700+900=3600\] \[\Rightarrow \,AB=60\,\,m\]. \[\cos \alpha =\frac{AD}{AB}=\frac{30\sqrt{3}}{60}=\frac{\sqrt{3}}{2}\] \[\Rightarrow \alpha ={{\tan }^{-1}}\left( \frac{\sqrt{3}}{2} \right)\]