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 π 5 , and it is changing at a rate of 0.2 rad / min . How fast is the balloon rising at this moment?

Answer 1

0.8 rad/ min

Explanation:

The height of the balloon at any point in time can be calculated using relationships of a right-angled triangle:

tan ϴ = h/r

h= r tan ϴ------------------------------------------------------------------------ (1)

dh/dt = (dh/dϴ) x dϴ/dt------------------------------------------------------- (2)

dh/dϴ (differentiating equation (1) = r Sec2 ϴ

dϴ/dt = 0.2 rad / min

r =4km

ϴ = 5 π

Substituting into (2), we have:

dh/dt = 4 Sec2 5 π x 0.2

= 4 (-1)2 x 0.2

= 4 x 0.2

= 0.8 rad/ min