Calc rates question

Question

A hot air balloon rising vertically is tracked by an observer located 2 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is π/5, 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

Let y is y-coordinate of the balloon. Then y = 2tanθ. THen speed v = dy/dt = 2sec²θdθ/dt = 2sec²π/5·01mi/min = 0.30557 mi/min.