What are the sizes of the tanks.
Suppose we have 3 tanks, A, B, and C that hold 200, 150, and 100 galons of brine, respectively. Also, suppose that the system is closed, and further that the liquid (which is well-stirred) is pumped between the tanks only.
Tank A is connected to B, and tank B is connected to C. Tank C is connected to A.
Mixture of 4 gal/min goes from tank A to tank B.
Mixture of 4 gal/min goes from tank B to tank C.
Mixture of 4 gal/min goes from tank C to tank A.
(So the flow goes A -> B -> C -> A)
1.) Construact a model for the number of pounds of salt x_1(t), x_2(t), x_3(t) at time t in the tanks A, B, C (respectively).
2.) From #1, write a system in the form X' = AX.
Okay so I will let be the volume of each of the three tanks.
Let be amount of salt in each each tank at any time .
The differencial equation for is:
.
The rate in is what is coming in from tank C. Now C is coming in at 4 gallons per minute, so we need to multiply this by its concentration which is so that works out to be . The rate out is what is going out of A. That is 4 gallons per minute times the concentration of salt in A which also is .
Thus,
.
Similarly,
.
.
To write in matrix form let,
.
And let,
.
So, (without initial conditions)