Chapter 8.3, Problem 11E

Consider the large mixing tanks shown in Figure 8.3.1. 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.

1. (a) Construct a mathematical model in the form of a linear system of first-order differential equations for the number of pounds x₁(t) and x₂(t) of salt in tanks A and B, respectively, at time t. Write the system in matrix form. [Hint: Review Section 3.3.]
2. (b) Use the method of undetermined coefficients to solve the linear system in part (a) subject to x₁(0) = 60, x₂(0) = 10.
3. (c) What are $\underset{t\to \infty }{\lim }{x}_1(t)$ and $\underset{t\to \infty }{\lim }{x}_2(t)$? Interpret this result.
4. (d) Use a graphing utility to plot the graphs of x₁(t) and x₂(t) on the same coordinate axes.

FIGURE 8.3.1 Mixing tanks in Problem 11