A hot air balloon rising vertically is tracked by an observer located 5 miles from the lift-off point.

Question

A hot air balloon rising vertically is tracked by an observer located 5 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is π/6 , and it is changing at a rate of 0.1 rad/min.

How fast is the balloon rising at this moment? ____miles/min

Answer 1

This is really a calculus problem involving the chain rule. Given the geometry presented, let R = the angle, and H = the height of the balloon. At any point in time, tan(R) = H/5. Differentiating both sides with respect to time and properly applying the chain rule: sec^2(R)*dR/dt = 0.2 dH/dt. dR/dt is given as 0.1 r/min, and R = pi/6. You can then solve for dH/dt which is how quickly the height of the ballon is changing.