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 =

Answer 1

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