Question

A hot air ballon rising vertically is tracked by an observer located 3 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?

Answer 1

0.458 mi/min

Explanation:

We are given that

Distance of observer from lift-off point=3 miles

`\theta=(\pi)/(5)`

`(d\theta)/(dt)=0.1rad/min`

We have to find the rate at which the balloon rising at this moment.

`tan\theta=(perpendicular\;side)/(base)`

Using the formula

`tan\theta=(h)/(3)`

Differentiate w.r.t t

`sec^2\theta(d\theta)/(dt)=(1)/(3)(dh)/(dt)`

Using the formula

`(d(tan\theta))/(d\theta)=sec^2\theta`

`(dh)/(dt)=3sec^2\theta(d\theta)/(dt)`

`(dh)/(dt)=3sec^2((\pi)/(5))* 0.1`

`(dh)/(dt)=0.458 mi/min`