Question

A man on the deck of a ship is 16 m above water level. He observes that the angle of elevation of the top of a cliff is 45° and the angle of depression of the base is 30°. Calculate the distance of the cliff from the ship and the height of the cliff.

Answer 1

Given:

A man on the deck of a ship is 16 m above water level.

Angle of elevation to the top of the cliff = 45°.

Angle of depression to the base of the cliff = 30°.

Step-wise calculation:

1. Let x = horizontal distance from the ship to the cliff and H = height of the cliff above water.

2. From the angle of depression to the base 30°: 

`tan 30^circ = "Vertical drop"/"Horizontal distance"`

= `16/x`

`tan 30^circ = 1/sqrt(3)`

So, `1/sqrt(3) = 16/x`

⇒ `x = 16sqrt(3)`

3. From the angle of elevation to the top 45°: 

`tan 45^circ = (H - 16)/x = 1`

So, H – 16 = x

⇒ H = x + 16

= `16sqrt(3) + 16` 

= `16(sqrt(3) + 1)`

Distance of the cliff from the ship = `16sqrt(3)` ≈ 27.71 m.

Height of the cliff = `16(sqrt(3) + 1)` ≈ 43.71 m.