$1.(10$ points$)$ Suppose that a tank originally contains $1$ kilogram of salt dissolved in $200$ liters of water. Then, fresh water is poured into the tank at a rate of $2$ liters per minute. The mixture gets stirred well in the tank and drains from the tank at the same rate of $2$ liters per minute. This is Stage I of the process. After $10$ minutes, Stage I terminates, and a mixture containing $0\mathrm{.}1$ kilograms of salt per liter is poured into the tank at a rate of $2$ liters per minute, with the mixture again leaving at the same rate. This is Stage II of the process.

$($a$)$ Set up two initial value problems that describe both stages of the process $($one initial value problem per stage$).$

$($b$)$ Solve both initial value problems and find the quantity of salt in the tank after $15$ minutes $(10$ minutes in Stage I and $5$ minutes in Stage II$).$

$($c$)$ Determine $\backslash (\backslash$lim $\_\{$t $\backslash$rightarrow $\backslash$infty$\}$ Q$($t$)\backslash ),$ where $\backslash ($ Q$($t$)\backslash )$ is the quantity of salt in the tank at time $\backslash ($ t $\backslash ).$