The air in a 49 cubic metre kitchen is initially clean, but when Elaine burns her toast while making breakfast, smoke is mixed with the room's air at a rate of 0.08 mg per second. An air conditioning system exchanges the mixture of air and smoke with clean air at a rate of 8 cubic metres per minute. Assume that the pollutants are mixed uniformly throughout the room and that burnt toast is taken outside after 31 seconds. Let S(t) be the amount of smoke in mg in the room at time t (in seconds) after the toast first began to burn.

1.Find a differential equation obeyed by S(t).

2.Find S(t) for 0≤t≤31 by solving the differential equation in (a) with an appropriate initial condition.

3.What is the level of pollution in mg per cubic meter after 31 seconds?

4.How long does it take for the level of pollution to fall to 0.008 mg per cubic metre after the toast is taken outside?