A Mixing Problem A pair of tanks are connecteed as shown in figure 3.3.12. At t = 0,tan...

A Mixing Problem A pair of tanks are connecteed as shown in figure 3.3.12. At t = 0,tank A contains 500 litres of liquid, 200 of which are ethanol ans tank B Contains 100 litres of liquid, 7 of which are ethanol.Begining at t – 0.3 litres of 20% ethanol solution arew added per minute. An additional 2 L/min are pumped from tank B back into tank A. The result is continously mixed, and 5 L/min are pumped into tank B.The contents of tank B are also continously mixed.In addditoon to the 2 litres they are returned to tank A, 3 L/min are discharged from the system.Let P(r) and Q(t) denote the number of litres of ethanol in tanks A and B at time t.We wish to find P(t).Using the pronciple that

(a) Qualitatively discuss the behaviour 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 t —> ∞?