# A mixing problem with concentrations

These mixing problems trip me up sometimes and I was just wondering if my setup was correct. It asks: A tank with a capacity of 500 gallons originally contains 200 gallons of water with 100 lb. of salt in solution. Water containing 1 lb. of salt per gallon is entering at the rate of 3 gallons per minute, and the mixture is allowed to flow out of the tank at a rate of 2 gallons per minute. Find the concentration $C(t)$ of salt in the tank at any time t until it overflows. So the set up for the Cauchy problem I had was this:

\begin{align*} C(0)&=\frac{1}{2}\\ \frac{dC}{dt}&=\frac{\text{rate of change of salt}}{\text{rate of change of fluid in tank}}\\ &=\frac{ \frac{1lb}{gal} \cdot \frac{3gal}{min}- \frac{2gal}{min}\cdot \frac{Clb}{gal}}{\frac{3gal}{min}-\frac{2gal}{min}}\\ &=\frac{\frac{3lb}{min}-\frac{2Clb}{min}}{\frac{1gal}{min}}\\ &= \frac{3lb}{gal}-\frac{2Clb}{gal}\\ &=3-2C \end{align*}

and from there the solution is just algebra.

• The intuition may be better if we find a DE for the amount $S(t)$ of salt in the tank at time $t$. – André Nicolas Aug 18 '15 at 18:18