D$2$a$.$

A $600$ gallon tank initially contains $200$ gallons of water and $20$ ltss of salt. There is an inflow to the tank at a rate of $\backslash (4\backslash$mathrm$\{$gal$\}/\backslash$mathrm$\{$min$\}\backslash )$ containing $\backslash (1\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$gal$\}\backslash )$ of salt. Water is removed from the well$-$mixed tank at a rate of $\backslash (2\backslash$mathrm$\{$gal$\}/\backslash$mathrm$\{$min$\}\backslash ).$

$($a$)$ Set up and solve an initial value problem for the volume $\backslash ($ V$($t$)\backslash )$ of water in the tank, and use this to set up an initial value problem $($including all of the necessary components$)$ for the amount of salt $\backslash ($ S$($t$)\backslash )$ in the tank.

$($b$)$ Solve this initial value problem for $\backslash ($ S$($t$)\backslash ).$

$($c$)$ Is there a point in time where this differential equation stops making sense for this system? If so$,$ determine that point in time, what happens at that point, and determine how much salt is in the tank at that moment.

$($d$)$ So$,$ there's going to be an issue here. In order to fix this, you are going to add a spill$-$over tank, which will collect everything running out of the top of the tank. We want to write a new set of initial value problems to model this situation with the two tanks. Using $\backslash ($ S$\_\{1\}($t$)\backslash )$

to represent the salt content in the initial tank, write an initial value problem for $\backslash ($ S$\_\{1\}\backslash )$ in this new situation. Since the tank is at capacity, the volume of water in the tank is constant. There are also two outflows from that tank. the one that flows out of the system and the one that goes into the spillover tank. This will be a differential equation that just involves $\backslash ($ S$\_\{1\}\backslash ).$ You should take the initial time to be the moment this tank overflows $($so the initial salt value is the value you just computed in the previous part$).$

$($e$)$ Next, we want to write a differential equation for the salt content in the spill$-$over tank $\backslash ($ S$\_\{2\}\backslash ).$ This tank has a capacity of $200$ gallous, but initially holds $100$ gallons of clean water. To this tank, you will add clean water at a rate of $\backslash (3\backslash$mathrm$\{$gal$\}/\backslash$mathrm$\{$min$\}\backslash )$ as well as the spill$-$over from the initial tank, and remove water at a rate to keep the volume of the spill$-$over tank at $100$ gallons. This tank is abo well$-$mixed. Set up an initial value problem, starting at the moment the first tank overflows, for $\backslash ($ S$\_\{2\}($t$)\backslash ).($Hint: This equation will involve $\backslash ($ S$\_\{1\}\backslash ),$ since one of the in$-$flow streams is coming from the initial tank, but that is fine. Tank $2$ also starts with no salt because it is clean water.$)$

$($f$)$ Solve the first initial value problem for $\backslash ($ S$\_\{1\}($t$)\backslash ).$ This one should involve exponentials because the volume in tank $1$ is constant.

$($g$)$ Take the solution you had for $\backslash ($ S$\_\{1\}($t$)\backslash ),$ plug this into the equation for $\backslash ($ S$\_\{2\}($t$)\backslash ),$ and then solve that initial value problem for $\backslash ($ S$\_\{2\}($t$)\backslash ).$

$($h$)$ Determine the long$-$term value for $\backslash ($ S$\_\{1\}($t$)\backslash )$ and $\backslash ($ S$\_\{2\}($t$)\backslash )$ and what this looks like in terms of concentration of the outflow stream from each tank. Do these values make sense?