If the angles of elevation of the top of a tower from three...

Let, the height of the tower$= h$

Let, $AB = x, BC = y, CD = z$

In $∆ ADE,$$tan {30}^{o} = \dfrac{h}{x + y + z}$

$⇒ x + y + z = \sqrt{3}h . . . \left(i\right)$

In $∆ BDE, \tan 45 ° = \dfrac{h}{y + z}$

$⇒ y + z = h . . . \left(ii\right)$

In $∆ CDE, \tan 60 ° = \dfrac{h}{z}$

$⇒ z = \dfrac{h}{\sqrt{3}} . . . \left(iii\right)$

From \left(ii\right) \text{ and } \left(iii\right),$<br /><br />$y = h\left(1 - \dfrac{1}{\sqrt{3}}\right) . . . \left(iv\right)$<br /><br />From \left(i\right) \text{ and } \left(ii\right),$

$x = \left(\sqrt{3} - 1\right)h . . . \left(v\right)$

$∴ \dfrac{AB}{BC} = \dfrac{x}{y} = \dfrac{\left(\sqrt{3} - 1\right)h}{h\left(\dfrac{\sqrt{3} - 1}{\sqrt{3}}\right)}$$= \dfrac{\sqrt{3}}{1}$