Solved: Consider the large mixing tanks shown in FIGURE 10.4.1. Suppose that both tanks

Chapter 10, Problem 11

(choose chapter or problem)

Consider the large mixing tanks shown in FIGURE 10.4.1. Suppose that both tanks A and B initially contain 100 gallons of brine. Liquid is pumped in and out of the tanks as indicated in the figure; the mixture pumped between and out of the tanks is assumed to be well-stirred.

(a) Construct a mathematical model in the form of a linear system of first-order differential equations for the number of pounds \(x_{1}(t)\) and \(x_{2}(t)\) of salt in tanks A and B, respectively, at time t. Write the system in matrix form. [Hint: See Section 2.9 and Problem 44, Chapter 2 in Review.]

(b) Use the method of undetermined coefficients to solve the linear system in part (a) subject to \(x_{1}(0)=60, x_{2}(0)=10\).

(c) What are \(\lim _{t \rightarrow \infty} x_{1}(t)\) and \(\lim _{t \rightarrow \infty} x_{2}(t)\)? Interpret this result.

(d) Use a graphing utility or CAS to plot the graphs of \(x_{1}(t)\) and \(x_{2}(t)\) in the same coordinate plane.