Bravo! That is the right answer.
The Equation of Continuity is
Let the subscripts "1" represent the duct and "2" represent the room. Even then, it looks like we might be in trouble for we don't know anything about v2 the velocity in the room and it's not at all clear what we would take as A2, some sort of cross-sectional area for the room. Nonetheless, let's start with this and see what happens.
A1 v1 = A2 ( l2/ t)
A1 v1 = (A2 l2) / t
A1 v1 = V2 / t
V2 is the total volume of the room and t is the time of 20 minutes,
V2 = (2.4 m) (3.0 m) (4.0 m)
V2 = 28.8 m3
A1 v1 = (28.8 m3) / (20 min)
A1 v1 = 1.44 (m3/min)
A1 = (10 cm) (30 cm) = 300 cm2
As always, be careful with the units! It is probably easier to keep everything in meters from the very beginning,
A1 = (0.10 m) (0.30 m) = 0.03 m2
A1 v1 = (0.03 m2) v1 = 1.44 (m3/min)
v1 = [1.44 (m3/min)] / [ 0.03 m2 ]
v1 = 48 m/min
That answer is fine but we often or usually express speeds like this in m/s or cm/s so let's go ahead and make that conversion,
v1 = 48 m/min [ min/60 s ]
v1 = 0.8 m/s
That is the speed of the air from the duct or vent.
(c) 2000, Doug Davis; all rights reserved.