An 8 gallon vat is full of pure water. At time t = 0 salt water is added to the vat through a pipe carrying water at a rate of 3 gallons per minute and a concentration of salt of 1/2 a pound per gallon. Water drains out of the vat at a rate of 3 gallon per minute, so that the level of the vat is always 6 gallons. Assume that the salt is always evenly mixed throughout the vat. Let S(t) denote the amount of salt in the vat at time t, and let t be measured in minutes.
a. Set up the differential equation and initial condition for dS/dt for the situation above.
b. Find S(t).