Angles of elevation of the top of a tower from three points (collinear) \(A, B\) and \(C\) on a road leading to the foot of the tower are \(30^{\circ}, 45^{\circ}\) and \(60^{\circ}\) respectively. The ratio of \(A B\) to \(B C\) is

By sine law, in \(\triangle A B Q\)
\(\frac{\sin 15^{\circ}}{A B}=\frac{\sin 30^{\circ}}{B Q}\)

\(\Rightarrow B Q=\frac{\sin 30^{\circ} \cdot A B}{\sin 15^{\circ}}\)
By sine law, in \(\triangle B C Q\)
\(\frac{\sin 120^{\circ}}{B Q}=\frac{\sin 15^{\circ}}{B C} \)
\(\Rightarrow B Q=\frac{B C \sin 120^{\circ}}{\sin 15^{\circ}}\)
From Eqs. (i) and (ii), we get
\(\frac{\sin 30^{\circ} \cdot A B}{\sin 15^{\circ}} =\frac{B C \cdot \sin 120^{\circ}}{\sin 15^{\circ}} \)
\(\Rightarrow \frac{A B}{B C}-\frac{\sin 120^{\circ}}{\sin 30^{\circ}} =\frac{\cos 30^{\circ}}{\sin 30^{\circ}}-\frac{\sqrt{3} / 2}{1 / 2} \)
\(A B: B C =\sqrt{3}: 1\)