Question

A man on deck of a ship is 10 m above the water level. He observes that the angle of elevation at 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 height of the cliff.

Answer 1

Given:

Height of man above water = 10 m point A.

Let D be the foot base of the cliff at water level and C the top of the cliff.

Let x = horizontal distance AD ship to cliff.

Let h = height of the cliff CD.

Step-wise calculation:

1. From A, angle of depression to D is 30°.

In the right triangle ADN horizontal AD = x, vertical drop AN = 10:

`tan 30^circ = "Opposite"/"Adjacent"`

= `10/x`

⇒ `x = 10/(tan 30^circ)`

= `10/(1/sqrt(3))`

= `10sqrt(3)`

= 17.32 m

2. From A, angle of elevation to C is 45°.

In the right triangle ANC vertical rise CN = h – 10, horizontal AN = x:

`tan 45^circ = (h - 10)/x = 1`

⇒ h – 10 = x

⇒ h = x + 10

= `10sqrt(3) + 10` 

= 27.32 m

Distance of the cliff from the ship = `10sqrt(3)` m ≈ 17.32 m.

Height of the cliff = `10(1 + sqrt(3))` m ≈ 27.32 m.