The air in a 83 cubic metre kitchen is initially clean, but when Paul burns his toast while making breakfast, smoke is mixed with the room's air at a rate of 0.07 mg per second. An air conditioning system exchanges the mixture of air and smoke with clean air at a rate of 7 cubic metres per minute. Assume that the pollutant is mixed uniformly throughout the room and that burnt toast is taken outside after 48 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. a. Find a differential equation obeyed by S(t). b. Find S(t) for 0