The angle of elevation of the summit of a mountain from a point on...

In$∆ CDF$

$\sin 30 ° = \dfrac{z}{1}\left[CD = 1km\right]$

(given)

$z = \dfrac{1}{2}$ ................(1)

$\cos 30 ° = \dfrac{y}{1} ⇒ \dfrac{\sqrt{3}}{2}$

now in

$∆ ABC$

$\tan 45 ° = \dfrac{h}{x + y}$

$⇒ h = x + y$

$⇒ x = h - \dfrac{\sqrt{3}}{2}$ .............(2)

now in

$∆ BDE,$

$\tan 60 ° = \dfrac{h - z}{x}$

$\sqrt{3}x = h - \dfrac{1}{2} ⇒ \sqrt{3}\left(h - \dfrac{\sqrt{3}}{2}\right) = h - \dfrac{1}{2} ⇒ \left(\sqrt{3} - 1\right)h = 1$

$h = \dfrac{1}{\sqrt{3} - 1}km$