Consider a 20-gallon vat that at time t = 0 contains 5 gallons of clean water. Suppose water enters the vat from two pipes. From the first pipe, salt water containing 2 pounds of salt per gallon enters the vat at a rate of 3 gallons per minute. From the second pipe, salt water containing 0.5 pounds of salt per gallon enters the vat at a rate of 4 gallons per minute. Suppose the water is kept well mixed and salt water is removed from the vat at a rate of 2 gallons per minute.
(a) Derive a differential equation for the rate of change of the total amount of salt S(t) in the vat at time t .
(b) Convert this equation to a differential equation for the concentration C(t) of salt in the vat at time t .
(c) Find the concentration of salt in the vat at the instant when the vat first starts to overflow.
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