Consider a reservoir maintaining a saltwater solution with concentration $C_0,$ which is

pumped into a tank of saltwater with volume $V$ at flow rate $F.$ The contents of this

tank are pumped out at flow rate $F$ into a second tank with volume $W.$ The contents

of the second tank are pumped out at flow rate $F.$

a$)$ Write a system of differential equations to describe the change in the concentration

of saltwater in both tanks, denoted A and $B$ respectively.

b$)$ Demonstrate the equations provided in the previous part lead to consistent units.

c$)$ Let $A=A^{**}$hat$(A),B=B^{**}$hat$(B),$ and $t=t^{**}$hat$(t)$ and substitute into the differential equations

you obtained to formulate expressions for $\frac{d{A}^{**}}{d{t}^{**}}$ and $\frac{d{B}^{**}}{d{t}^{**}}.$

d$)$ Let hat$(t)=\frac{W}{F}$ and hat$(A)=$hat$(B)=\frac{W{C}_0}{V}.$ What system of differential equations is obtained?

e$)$ What parameters result from the above substitution?

f$)$ What are the physical interpretations of the units and parameters?

g$)$ What are the steady$-$state solutions for the resulting system?

h$)$ What is the physical interpretation of each steady state?