question_answer

\[\angle BCA=\beta =\]

A) \[{{\tan }^{-1}}\left( \frac{1}{2} \right)\]

B) \[{{\tan }^{-1}}\left( 2 \right)\]

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

D) \[{{\tan }^{-1}}\left( \sqrt{3} \right)\]

Correct Answer: D

Solution :

\[A{{B}^{2}}={{\left( AD \right)}^{2}}+{{\left( BD \right)}^{2}}={{\left( 30\sqrt{3} \right)}^{2}}+{{\left( 30 \right)}^{2}}\] \[=2700+900=3600\] \[\Rightarrow \,AB=60\,\,m\]. \[\tan \beta =\frac{BD}{DC}=\frac{30}{10\sqrt{3}}=\frac{3}{\sqrt{3}}=\sqrt{3}\] \[\Rightarrow \,\,\beta ={{\tan }^{-1}}\left( \sqrt{3} \right)\]