A tank initially contains $100 \text{gal}$ of brine, a solution of salt and water, whose salt concentration is $0.5 \frac{\text{lb}}{\text{gal}}$. Brine whose salt concentration is $2 \frac{\text{lb}}{\text{gal}}$ flows into the tank at the rate of $3\frac{ \text{gal}}{ \text{min}}$. The mixture flows out at the rate of $2\frac{ \text{gal}}{ \text{min}}$. Assume the salt is uniformly distributed throughout the mixture. Find the salt concentration of the brine at the end of $30$ minutes.
Letting $x $ be the amount of salt at any time $t$
I got
$$
\frac{\text{dx}}{\text{dt}}=6-\frac{2x}{100+t}
$$
$$
x = \frac{6[10,000t+100t^2+\frac{t^3}{3}]}{(100+t)^2}
$$
What is the meaning of concentration of the brine? Is it $\frac{\text{dx}}{\text{dt}}$?
Best Answer
(NOTE: My first answer was wrong. This is a corrected answer.)
Concentration is the amount of solute (salt in this case) divided by the volume of the solution (brine in this case). So assuming that the unit of $x$, the amount of salt, is pounds, the concentration of the brine in the tank at the start is
$$\frac x{100}\frac{\text{lb}}{\text{gal}}$$
However, the volume in the tank does not remain at $100$ gallons. The inflow of liquid has the rate $3\frac{\text{lb}}{\text{gal}}$ while the outflow has the rate $2 \frac{\text{lb}}{\text{gal}}$, so the net inflow is $1 \frac{\text{lb}}{\text{gal}}$. So the volume of the tank at time $t$ is not $100$ but rather $100+t$.
Therefore the concentration of salt in the tank at time $t$ is
$$\frac x{100+t}\frac{\text{lb}}{\text{gal}}$$
Because of that, your differential equation is correct. The rate of flow of salt into the tank is $2 \frac{\text{lb}}{\text{gal}}\cdot 3\frac{ \text{gal}}{ \text{min}}=6\frac{\text{lb}}{\text{min}}$. The rate of flow of salt out of the tank is the concentration of the brine times the rate of flow out, namely $\frac{x}{100+t}\cdot 2\frac{ \text{lb}}{ \text{min}}$. Therefore, your equation is
$$\frac{dx}{dt}=6-\frac{2x}{100+t}$$
with the initial value
$$x(0)=100\cdot 0.5$$
How did you get the correct equation without understanding the concentration? Anyway, the meaning of $\frac{dx}{dt}$ is the rate of change of the mass of salt in the tank, not the concentration.