A 250 gallon tank initially contains200 gallonsof pure water. At timet = 0a pipe starts carrying a salt-water solution into the tank at a rate of20.0 gal/h. The water flowing into the tank has a0.100 lb/galconcentration 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 of20.2 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.

I have done this already:

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

$\displaystyle V(t) = 200 - 0.2t$

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.

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

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

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

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

Is this correct?