Chapter 1.7, Problem 7P

A tank initially contains w liters of a solution in which are dissolved $A_0$ grams of chemical. A solution containing $k\text{g/L}$ of this chemical flows into the tank at a rate of $r\text{L}/\text{min}$, and the mixture flows out at the same rate.

(a) Show that the amount of chemical, $A\left(t\right)$, in the tank at time t is

$A\left(t\right)=e^{-\left(rt\right)/w}\left[kw\left(e^{\left(rt\right)/w}-1\right)+A_0\right]$

(b) Show that as $t\to \infty$, the concentration of chemical in the tank approaches $k\text{g/L}$. Is this result reasonable? Explain.