question_answer

The angles of elevation of the top of a tower from the top and bottom at a building of height a are \[30{}^\circ \] and \[45{}^\circ \] 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) \[a(\sqrt{3}-1)\]

Correct Answer: C

Solution :

In \[\Delta ABC,\] \[\tan 30{}^\circ =\frac{AC}{BC}\] or \[\frac{1}{\sqrt{3}}=\frac{x}{BC}\]

or \[BC=x\sqrt{3}\]and in \[\Delta ADE,\] \[\tan 45{}^\circ =\frac{a+x}{DE}\]

or \[1=\frac{a+x}{x\sqrt{3}}\]  or \[x\sqrt{3}=a+x,\]\[\Rightarrow \,x(\sqrt{3}-1)=a\]

or         \[x=\frac{a}{\sqrt{3}-1}\]

Therefore, height of the tower,

\[a+x=a+\frac{a}{\sqrt{3}-1}\]

\[=a\left[ \frac{\sqrt{3}-1+1}{\sqrt{3}-1} \right]=\frac{a\sqrt{3}}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}=\frac{a(3+\sqrt{3})}{2}\]