$4.(10$ marks$)$ Consider the mixing process shown in the figure. A mixing chamber initially contains $5$ 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 $0\mathrm{.}25$ 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. $($b$)$ The objective is to obtain a dye concentration in the outflow mixture of $0\mathrm{.}2$ kilograms per liter. What injection rate $\backslash ($ r $\backslash )$ is required to achieve this equilibrium solution? $($c$)$ 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? $($d$)$ Assume the mixing chamber contains $5$ liters of clear liquid at time $\backslash ($ t$=0\backslash ).$ How many minutes will it take for the outflow concentration to rise to within $\backslash (4\backslash \%\backslash )$ of the desired concentration of $0\mathrm{.}2$ kilograms per liter?