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 π/5, and it is changing at a rate of 0.4 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).

1.22 km/min

Step-by-step explanation:

Let Q be baloon height at a time t. Our goal is to determine the speed of the baloon at the moment.

The dy / dt velocity of the baloon when = π/5?  

So we can restate the question as follows:

Owing to the fact d / dt = 0.2 rad / min at some stage = π/5

from fig:

tanθ = y/4

Differentiating w.r.t "t"

sec2 θ * dθ/dt = 1/4(dy/dt)

=> dy/dt = (4/cos2 θ)dθ/dt

At the given moment θ =  and dθ/dt = 0.2 rad/min.

dy/dt = (4/cos2)* (0.2)

= 1.22 km/min

And the velocity of the baloon currently is 1.22 km / min.

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