# Thread: Three Tanks

1. ## Three 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.

2. What are the sizes of the tanks.

3. Originally Posted by ThePerfectHacker
What are the sizes of the tanks.
The tanks are all the same size.

4. Okay so I will let $V$ be the volume of each of the three tanks.
Let $A(t),B(t),C(t)$ be amount of salt in each each tank at any time $t$.
The differencial equation for $A(t)$ is:
$\frac{dA}{dt} = \mbox{rate in } - \mbox{ rate out}$.
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 $B/V$ so that works out to be $4B/V$. 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 $4A/V$.
Thus,
$\frac{dA}{dt} = \frac{4}{V}C - \frac{4}{V}A$.
Similarly,
$\frac{dB}{dt} = \frac{4}{V}A - \frac{4}{V}B$.
$\frac{dC}{dt} = \frac{4}{V}B - \frac{4}{V}C$.
To write in matrix form let,
$\bold{X}(t) = \left( \begin{array}{c}A(t)\\B(t)\\C(t) \end{array}\right)$.
And let,
$A = \frac{4}{V} \left( \begin{array}{ccc} -1&0&1\\1&-1&0\\0&1&-1 \end{array} \right)$.

So, (without initial conditions)
$\bold{X}' = A\bold{X}$

5. Suppose we have 3 tanks, A, B, and C that hold 200, 150, and 100
gallons of brine, respectively.

If they're all the same size, then what does this mean?.

6. ## Three Tanks

Warning! Completely frivolous post ahead!

I must be getting tired because I can't resist doing this.

You're welcome, you're welcome, you're welcome!

-Dan