$1\mathrm{.}4$ Linear Equations: Problem $17$

$(1$ point$)$

Consider the mixing process shown in the figure. A mixing chamber initially contains $4$ 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 $0\mathrm{.}6$ kilograms per liter is injected into the mixing chamber at a constant rate of $r$ liters per minute. When the mixing process is started, the well$-$stirred mixture is pumped from the chamber at a rate of $10+r$ liters per minute.

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

$\frac{dQ}{dt}=k\frac{g}{m}in$

$($b$)$ The objective is to obtain a dye concentration in the outflow mixture of $0\mathrm{.}25$ kilograms per liter. What injection rate $r$ is required to achieve this equilibrium solution?

$r=$ L$/$min

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

$($c$)$ Assume the mixing chamber contains $4$ liters of clear liquid at time $t=0.$ How many minutes will it take for the outflow concentration to rise to within $5\%$ of the desired concentration of $0\mathrm{.}25$ kilograms per liter?

$t=$ min