The cliff of a mountain is 180 m high, and the angles of depression of two ships on the same side of cliff are 30° and 60°. Find the distance between the two ships in meter?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(120\sqrt{3}\)
The angle of depression of a ship seen from the top of the cliff is equal to the angle of elevation of the top seen from the ship (alternate angles between the two horizontal lines). In the right triangle formed by the cliff and the line of sight, \(\tan\theta=\frac{\text{height}}{\text{distance from the foot}}\), so the distance is the height divided by \(\tan\theta\).
Distance of each ship from the foot B of the cliff (height AB = 180 m):
Nearer ship C, angle 60°: \(\displaystyle BC=\frac{180}{\tan60^\circ}=\frac{180}{\sqrt{3}}=60\sqrt{3}\) m
Farther ship D, angle 30°: \(\displaystyle BD=\frac{180}{\tan30^\circ}=180\sqrt{3}\) m
Both ships are on the same side of the cliff, so the distance between them is the difference: \(CD=180\sqrt{3}-60\sqrt{3}=120\sqrt{3}\) m.
