Answer: 1318 Two-compartment mixing problems are similar to the mixing problems in

Chapter 10, Problem 13

(choose chapter or problem)

1318 Two-compartment mixing problems are similar to the mixing problems in Exercises 7.4.4548, except there are two connected tanks of fixed capacities filled with a well-mixed solution of some substance (for example, salt). Solution of a given concentration can enter each tank at a fixed rate, and solution also flows back and forth between the tanks. Solution can also leave each tank at a fixed rate. fc fc d d k k V V Using x1std and x2std to denote the amount of substance in each tank at time t, we obtain the following system of linear differential equations: dx1 dt f1c1 2 k12 x1 V1 1 k21 x2 V2 2 d1 x1 V1 dx2 dt f2c2 2 k21 x2 V2 1 k12 x1 V1 2 d2 x2 V2 Two tanks containing a salt solution are connected as in thefigure:r crVV r Suppose that x1std and x2std denote the amount of salt (ingrams) in each tank at time t (in minutes), and r 8 Lymin,c 2 gyL, V1 50 L, and V2 25 L. The initial amountsof salt in each tank are x1s0d 8 g and x2s0d 0 g. (a) What is the initial-value problem describing thissystem? (b) Solve the initial-value problem from part (a). (c) What is the maximum amount of salt that is ever intank2?