A lecture theatre having volume 1000m³ is designed to seat 200 people. The air is conditioned continuously by an inflow of fresh air at a constant rate V (in m³min^–1). An average person generates 980cm³ of CO₂ per minute, while fresh air contains 0.04% of CO₂ by volume. Show that the percentage concentration x of CO₂ by volume in the lecture theatre at time t (in min) after the audience enters satisfies the differential equation

If initially x(0) = 0.04, show that x(t) is an increasing function of t for t > 0. Deduce that the maximum x* of x (t) is given by

If the specification is that x does not exceed 0.06 (that is, 50% increase above fresh air), deduce that V must be chosen so that V > 980. Comment on this result.

Transcribed Image Text:

dx dt 1000- 19.6+0.04V Vx(t)