From the summit of a 100 m high cliff, the angles of depression to two boats situated on the same side of the cliff are measured at 30° and 45° respectively. The boats are moving towards the cliff in a straight line. After 5 minutes, the angle of depression to the first boat becomes 60°, while the angle to the second boat becomes 75°. If both boats move at constant but different speeds, determine the initial distance between the two boats.
Show answer & explanation
Correct answer: A
Explanation
Height of the cliff = 100 m
Angle of depression = Angle of elevation
tan θ = Height / Horizontal distance
First boat (initially 30°) -
tan 30° = 100 / d₁
1/√3 = 100 / d₁
d₁ = 100√3
≈ 173.2 m
Second boat (initially 45°) -
tan 45° = 100 / d₂
1 = 100 / d₂
d₂ = 100 m
Initial distance = d₁−d₂
= 173.2 – 100 = 73.2 m