A hot air balloon rising vertically is tracked by an observer located 4 km from the lift‑off point. At a certain moment, the angle between the observer's line of sight and the horizontal is 𝜋5, and it is changing at a rate of 0.1 rad/min. How fast is the balloon rising at this moment?

Question

A hot air balloon rising vertically is tracked by an observer located  4 km  from the lift‑off point. At a certain moment, the angle between the observer's line of sight and the horizontal is  𝜋5,  and it is changing at a rate of  0.1 rad/min.  How fast is the balloon rising at this moment?

Let  𝑦  be the height of the balloon (in km),  𝑡  be time (in minutes), and  𝜃  the angle between the line‑of‑sight and the horizontal (in radians).

(Use decimal notation. Give your answer to two decimal places.)

𝑑𝑦𝑑𝑡 =

oobleck · Answer

y/4 = tanθ 1/4 dy/dt = sec^2θ dθ/dt so plug and chug