Let A and B be two towers with the same base. From the mid point of the line joining their feet, the angles of elevation of the tops of A and B are 30° and 45°, respectively. The ratio of the heights of A and B is :

1: $\sqrt{3}$

In triangle RTS

⇒ tan\({30}^\circ\) = \(\frac{RS}{ST}\)

⇒ \(\frac{1}{√3}\) = \(\frac{RS}{ST}\)

⇒ RS = \(\frac{ST}{√3}\)

In triangle PTQ

⇒ tan\({45}^\circ\) = \(\frac{PQ}{QT}\)

⇒ 1 = \(\frac{PQ}{QT}\)3

⇒ PQ = QT

As QT = ST

⇒ √3 RS = PQ = A : B = 1 : √3

Therefore, ratio of towers A and B is 1 : √3.