At a certain moment a 100-gallon mixing tank is full of brine containing 0.25 pounds of salt per gallon.

Question

At a certain moment a 100-gallon mixing tank is full of brine containing 0.25 pounds of salt per gallon. Find the amount of salt present t minutes later if the brine is being continuously drawn off at the rate of 3 gallons per minute and replaced by brine containing 0.2 pounds of salt per gallon.

Answer 1

Let M = mass of salt in the tank as function of time

Note that dM/dt = rate of mass in -rate of mass out = 3*.2-3*M/100

Re-arrange the above to get dM/dt+.03M=.6, an ODE solved to find M(t)=20+Ce^-.03t where C is a constant of integration. Use the initial condition M(0)=.25*100=25 pounds to find C= 5pounds.