#### Question

A man standing on the deck of a ship, which is 10 m above water level, observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of a hill as 30°. Find the distance of the hill from the ship and the height of the hill

#### Solution

Let CD be the hill and suppose the man is standing on the deck of a ship at point A.

The angle of depression of the base C of the hill CD observed from A is 30° and the angle of elevation of the top D of the hill CD observed from A is 60°.

∴ ∠EAD = 60° and ∠BCA = 30°

In ΔAED,

tan60° = `(DE)/(EA)`

`:.sqrt3=h/x`

`:.h=sqrt3x `

In ABC

tan30° = `(AB)/(BC)`

`:.1/sqrt3=10/x`

`:.x=10sqrt3 `

Substituting x = 10 `sqrt3` in equation (1) we get

`h=sqrt3xx10sqrt3=10xx3=30`

∴ DE = 30 m

∴ CD = CE + ED = 10 + 30 = 40 m

Thus, the distance of the hill from the ship is `10sqrt3` m and the height of the hill is 40 m