# Math Problems Involving Mixing

1. All of the mixing problems that we will be dealing with will involve a “tank”.(mixing fluids in a tank)

Problem: Consider a tank which initially holds V0 gal of brine that contains a lb of salt. Another brine solution containing b lb of salt/gallon is poured into the tank at rate of gal/min, while simultaneously, the well-stirred solution leaves the tank at a rate of f gal/min

Objective: Find the amount of salt in the tank at any time t

2. The main assumption that we’ll be using here is that the concentration of the substance in the liquid is uniform throughout the tank. (in general the concentration is not the same) 3. A(t) is measuring the amount of substance , not the volume of the whole mixture, which is present in the tank at time t. Assumptions:

* rin = the rate at which the solution pours into the tank. * rout= the rate at which the mixture pours out of the tank * Cin= the concentration of salt in the solution being poured into the tank. * Cout= the concentration of salt in the solution being poured out of the tank. Remarks:

• If rin=rout then the “water level” of the tank stays the same. where V0denotes the volume of solution in the tank at time t. dAdt=Rin-Rout=rinCin-routCout

Cout=A(t)V0+rin-routt

Mixing Problems:

A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5gal/min. The well mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t. What is the concentration of the solution in the tank at t=5 min?

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