# Thread: First Order DE; Mixture Problem

1. ## First Order DE; Mixture Problem

A 250 gallon tank initially contains 200 gallons of pure water. At time t = 0 a pipe starts carrying a salt-water solution into the tank at a rate of 20.0 gal/h. The water flowing into the tank has a 0.100 lb/gal concentration of salt dissolved in the water. The water in the tank is constantly being stirred so the solution in the tank is at all times uniform. A second pipe carries the well-mixed water solution out of the tank at a rate of 20.2 gal/h. Let S(t) denote the number of pounds of salt dissolved in the water in the tank at the time t, and let V(t) denote the volume of the water solution in the tank at the time t.

a) Find the equation for the volume of the tank as a function of time.

$\frac{dV}{dt} = \frac{dV}{dt}|_{IN} - \frac{dV}{dt}|_{OUT}$

$V(t) = 200 - 0.2t$

b) Find the equation for the salt content as a function of time, S(t), given the boundary condition implied above.

This is the one I'm having trouble with. I just need confirmation on the differential equation.

$\frac{dS}{dt} = \frac{dS}{dt}|_{IN} - \frac{dS}{dt}|_{OUT}$

$\frac{dS}{dt}|_{IN} = 0.100 * 20.0 = 2 \frac{lbs}{h}$

$\frac{dS}{dt}|_{OUT} = \frac{S(t)}{V(t)}20.2 = \frac{20.2*S(t)}{200 - 0.2t}$

$\frac{dS}{dt} = 2 - 20.2\frac{S(t)}{200 - 0.2t}$

Is this correct?

2. Yep. That's what I got.