Question

A hot air balloon rising vertically is tracked by an observer located 4 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is pi/6, and it is changing at a rate of 0.1 rad/min. How fast is the balloon rising at this moment?

Answer 1

0.533 miles/min

Explanation:

Calculate how fast the balloon is rising at the moment : dy /dt

location from the lift-off point = 4 miles

angle between the observer's line-of-sight and the horizontal ( ∅ ) =
`(\pi )/(6)`

d∅ / dt = 0.1 rad/min

tan ∅ = y / 4

derivate

sec^2 ∅ . d∅ / dt = 1/4 . dy/dt

= sec^2
`(\pi )/(6)` . 0.1 = 1/4 . dy/dt

= ( sec^2 0.5238 * 0.1 ) * 4 = dy/dt

dy/dt = 8 / 15 miles/min