a man standing on deck of ship which is 10m above the sea level, observes the angle of elevation of the top of a cloud as 30 degree and angle of depression of its reflection in the sea was found to be 60 degree. find the height of the cloud and also the distance of cloud from the ship

Let AE be the height of the cloud and BD = 10m be the height of deck.

CE = BD = 10m

Let AC = h m

∴AE = AC + CE

= (h + 10) m

Now height of shadow is same

∴ AE = A' E

A' E = (h + 10) m

Now in ΔABC.

In ΔA' BC

⇒ h + 20 = 3h

⇒20 = 3h – h

⇒2h = 20

⇒h = 10 m

∴ height of the cloud = (h + 10) m

= (10 + 10) m

= 20 m

Put the value of h = 10 m in (1)

$\mathrm{BC} = 10\sqrt{3} \mathrm{m}$

Now in ΔABC, applying Pythagoras theorem

${\mathrm{AB}}^2 = {\mathrm{BC}}^2 + {\mathrm{AC}}^2$

$$\begin{gathered} {\mathrm{AB}}^2 = {\left(10\sqrt{3}\right)}^2 + {10}^2 \\ {\mathrm{AB}}^2 = 300 + 100 \\ {\mathrm{AB}}^2 = 400 \\ \mathrm{AB} = 20 \mathrm{m} \end{gathered}$$

Hence the height of the cloud is 20 m and the distance between cloud and ship is 20 m.