A Mixing Problem A pair of tanks are connected as shown in Figure 3.3.12. At t = 0,...

A Mixing Problem A pair of tanks are connected as shown in Figure 3.3.12. At t = 0, tank A contains 500 liters of liquid, 200 of which are ethanol, and tank B contains 100 liters of liquid, 7 of which are ethanol. Beginning at t = 0, 3 liters of 20% ethanol solution are added per minute. An additional 2 L/min are pumped from tank B back into tank A. The result is continuously mixed, and 5 L/min are pumped into tank B. The contents of tank B are also continuously mixed. In addition to the 2 liters that are returned to tank A, 3 L/min are discharged from the system. Let P(t) and Q(t) denote the number of liters of ethanol in tanks A and B at time t. We wish to find P(t). Using the principle that

rate of change = input rate of ethanol - output rate of ethanol,

we obtain the system of first-order differential equations

(a) Qualitatively discuss the behavior of the system. What is happening in the short term? What happens in the long term?

(b) We now attempt to solve this system. When (19) is differentiated with respect to t, we obtain

Substitute (20) into this equation and simplify.

(c) Show that when we solve (19) for Q and substitute it into our answer in part (b), we obtain

(d) We are given that P(0) = 200. Show that Then solve the differential equation in part (c) subject to these initial conditions.

(e) Substitute the solution of part (d) back into (19) and solve for Q(t).

(f) What happens to P(t) and Q(t) as