## Mixing problems, Mathematics

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In these problems we will begin with a substance which is dissolved in a liquid. Liquid will be entering as well as leaving a holding tank. The liquid entering the tank may or may not hold more of the substance dissolved into it. Liquid leaving the tank will of course comprise the substance dissolved in it. If Q (t) provides the amount of the substance dissolved into the liquid in the tank at any time t we need to develop a differential equation that, as solved, will provide us an expression for Q(t). Remember as well that in several situations we can think of air as a liquid for the reasons of these kinds of discussions and thus we don't actually require having an actual liquid, though could instead use air like the "liquid".

The major assumption that we'll be using here is which the concentration of the substance in the liquid is uniform during the tank. Obviously it will not be the case, although if we permit the concentration to vary depending upon the location into the tank the problem turns into very difficult and will include partial differential equations that are not the focus of this course.

The most important "equation" which we'll be using to model this situation is as:

Rate of change of Q(t)   = Rate at that Q(t) enters the tank - Rate at that Q(t) exits the tank

Here,

Rate of change of Q(t) = dQ/dt = Q'(t)

Rate at that Q(t) enters the tank= (flow rate of liquid entering) x (concentration of substance in liquid entering

Rate at that Q(t) exits the tank = (flow rate of liquid exiting) x (concentration of substance in liquid exiting)

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