Question

3. Consider a tank which initially contains V litres of water and Qo kilograms of salt. Suppose that a new mixture of brine at a concentration of k kg per litre is poured into the tank, the contents of the vat are thoroughly mixed, and the contents of the tank are drained at the same rate. Unlike the model we studied in class, however, now assume that the rate of inflow and outflow is proportional to the length of time since the mixing began (a) Set up a new differential equation to model this situation (b) Solve the equation, and determine what effect (if any) this change to the rate of in- flow/outflow has on the long-term behaviour of the salt concentration in the tank.

Answer 1

One tank with Capacity -V litres. Initially Qo kgs of salt is dissolved Let Q(t) be the kg of salt in the tank at time t. Thus Q(0)= Qo we form the differential equation To find Q(t) Suppose the rate of mixture of brine entering rate of contents draining = (1/min) r dQ rate of salt coming in - rate of salt going out = dt Q (t) kg kg l X r min tk- min Q = tkr V (4) Thus dQ rt (k dt b) dQ rtdt (k-9)

dQ rt dt V (Q kV) Integrating we get t2 +Const 2V n(QkV) Ce 2t2 (Q kV) Q(t) kV Ce 2vt 0,Q (t) Qo, thus As at t кV + C or,C - Qо — kV _ Thus the particular solution is Q (t) kVQo - kV)e zv As time passes, that is in the long run the term(Qo - kV)e quantity of salt in the tank will be kV (kg) of salt in the tank. vanishes. Thus in the long run the