From the top of a cliff 60 m high, the angles of depression of two boats are 30° and 60° respectively. Find the distance between the boats, when the boats are:

(i) on the same side of the cliff,

(ii) on the opposite sides of the cliff.

(i) Let R be the top of the cliff and Q be the foot of the cliff such that RQ = 60 m.

Let P and T be the positions of the two boats such that the angles of depression from R are 30° and 60° respectively.

From figure,

∠RPQ = 60° and ∠RTQ = 30°

Boats on the same side of the cliff

From right angled ΔTQR, we get

$\Rightarrow \tan \theta = \dfrac{\text{perpendicular}}{\text{base}} \\[1em] \Rightarrow \tan 30^{\circ} = \dfrac{RQ}{TQ} \\[1em] \Rightarrow \dfrac{1}{\sqrt{3}} = \dfrac{60}{TQ} \\[1em] \Rightarrow TQ = 60\sqrt{3}$

From right angled ΔPQR, we get

$\Rightarrow \tan \theta = \dfrac{\text{perpendicular}}{\text{base}} \\[1em] \Rightarrow \tan 60^{\circ} = \dfrac{RQ}{PQ} \\[1em] \Rightarrow \sqrt{3} = \dfrac{60}{PQ} \\[1em] \Rightarrow PQ = \dfrac{60}{\sqrt{3}}$

Distance between the boats,

$\Rightarrow PT = TQ - PQ \\[1em] \Rightarrow PT = 60\sqrt{3} - \dfrac{60}{\sqrt{3}} \\[1em] \Rightarrow PT = \dfrac{180 - 60}{\sqrt{3}} \\[1em] \Rightarrow PT = \dfrac{120}{\sqrt{3}} \\[1em] \Rightarrow PT = 40\sqrt{3} \\[1em] \Rightarrow PT = 69.28 \text{ m.}$

Hence, the distance between the boats is 69.28 m when they are on the same side of the cliff

(ii) Let R be the top of the cliff and Q be the foot of the cliff such that RQ = 60 m.

Let P and T be the positions of the two boats on opposite sides of cliff, such that the angles of depression from R are 30° and 60° respectively.

From figure,

∠RPQ = 60° and ∠RTQ = 30°

The distance between the boats, when they are on opposite sides of cliff,

$\Rightarrow PT = TQ + PQ \\[1em] \Rightarrow PT = 60\sqrt{3} + \dfrac{60}{\sqrt{3}} \\[1em] \Rightarrow PT = \dfrac{180 + 60}{\sqrt{3}} \\[1em] \Rightarrow PT = \dfrac{240}{\sqrt{3}} \\[1em] \Rightarrow PT = 80\sqrt{3} \\[1em] \Rightarrow PT = 138.56 \text{ m.}$

Hence, boats are 138.56 m when they are on the opposite sides of the cliff.