question_answer

The angle of elevation of the top of a tower A from the top B and bottom D at a building of height a are \[{{30}^{o}}\] and\[{{45}^{o}}\]respectively. If the tower and the building stand at the same level, then the height of the tower is:

A)  \[a\sqrt{3}\]                     

B)  \[\frac{a\sqrt{3}}{\sqrt{3}-1}\]

C)  \[\frac{a(3+\sqrt{3})}{2}\]                          

D)  none of these

Correct Answer: C

Solution :

\[{{\tan }^{{{30}^{o}}}}=\frac{AC}{BC}\] \[\frac{1}{\sqrt{3}}=\frac{x}{BC}\Rightarrow BC=\sqrt{3}x\] \[\tan {{45}^{o}}=\frac{x+a}{DE}\Rightarrow DE=x+a\Rightarrow \sqrt{3}x=a+x\] \[(\sqrt{3}-1)x=a\Rightarrow x=\frac{a}{\sqrt{3}-1}\] Height of the tower \[a+x=a+\frac{a}{\sqrt{3}-1}=a\left[ \frac{\sqrt{3}-1+1}{\sqrt{3}-1} \right]\] \[=\frac{a\sqrt{3}}{\sqrt{3}-1}\times \frac{\sqrt{3}\times 1}{\sqrt{3}+1}\] \[=\frac{a(3+\sqrt{3})}{2}\]