A hot-air balloon directly above point A is rising vertically at 2 m/s. An observer at point B, 500 m due east of A, watches the balloon. At the moment when the angle of elevation is 30°, how long until it reaches an angle of elevation of 60°? (Assume no horizontal drift.)
Show answer & explanation
Correct answer: A
Explanation
Let the height of the balloon = h meters
tan θ = Height / Base
tan θ = h/500
Angle of elevation = 30°
tan 30° = h₁/500
1/√3 = h₁/500
h₁ = 500/√3
≈ 288.675 m
Angle of elevation = 60°
tan 60° = h₂/500
√3 = h₂/500
h₂ = 500√3
≈ 866.025 m
Increase in height -
Δh = h₂−h₁
= 500√3 − 500/√3
= 1000/√3
≈ 577.35 m
Speed of the balloon = 2 m/s
Time = 577.35 / 2 = 288.675 seconds