$(1$ point$)$

Consider the mixing process shown in the figure. A mixing chamber initially contains $2$ liters of a clear liquid. Clear liquid flows into the chamber at a rate of $10$ liters per minute. A dye solution having a concentration of $0\mathrm{.}4$ kilograms per liter is injected into the mixing chamber at a constant rate of $\backslash ($ r $\backslash )$ liters per minute. When the mixing process is started, the well$-$stirred mixture is pumped from the chamber at a rate of $\backslash (10+$r $\backslash )$ liters per minute.

$($a$)$ Develop a mathematical model for the mixing process. Let $\backslash ($ Q $\backslash )$ represent the amount of dye in kilograms in the mixture.

$\backslash [$

$\backslash$frac$\{$d Q$\}\{$d t$\}=\backslash$quad $\backslash$quad $\backslash$mathrm$\{$kg$\}/\backslash$min

$\backslash ]$

$($b$)$ The objective is to obtain a dye concentration in the outflow mixture of $0\mathrm{.}1$ kilograms per liter. What injection rate $\backslash ($ r $\backslash )$ is required to achieve this equilibrium solution?

$\backslash ($ r$=\backslash )\backslash (\backslash$mathrm$\{$L$\}/\backslash$mathrm$\{$min$\}\backslash )$

Would this equilibrium value of $\backslash ($ r $\backslash )$ be different if the fluid in the chamber at time $\backslash ($ t$=0\backslash )$ contained some dye?

$($c$)$ Assume the mixing chamber contains $2$ liters of clear liquid at time $\backslash (\backslash$boldsymbol$\{$t$\}=0\backslash ).$ How many minutes will it take for the outflow concentration to rise to within $\backslash (3\backslash \%\backslash )$ of the desired concentration of $0\mathrm{.}1$ kilograms per liter?

$\backslash ($ t$=\backslash )\backslash (\backslash$min $\backslash )$