Ahot 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 π3 , and it is changing at a rate of 0.1 rad/min. how fast is the balloon rising at this moment?

1.6 miles/min..

Explanation:

Let y be the height of the balloon at time t. Our goal is to compute the balloon's velocity at the moment .

balloon's velocity dy/dt when θ =π/3 radian

so we can restate the problem as follows:

Given dθ/dt = 0.1 rad/min at θ = π/3

from the figure in the attachment

tanθ = y/5

Differentiating w.r.t "t"

sec^2 θ×dθ/dt = 1/4(dy/dt)

⇒ dy/dt = (4/cos^2 θ)dθ/dt

At the given moment θ =π/3 and dθ/dt = 0.1 rad/min.

therefore putting the value we get

solving we get

= 1.6 miles/min

So the balloon's velocity at this moment is 1.6 miles/min.