Differential Equation Involving Brine Solution in the Tank

(a) A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t.

(b) In part a, what is the concentration c(t) of the salt in the tank at time t? At t=5 min? What is the concentration of the salt in the tank after a long time, that is, as t→∞? At what time is the concentration of the salt in the tank equal to one-half this limiting value?

(c) Solve part a under the assumption that the solution is pumped out at a faster rate of 10 gal/min. When is the tank empty?