Two very large tanks A and B are each partially filled with 200 gallons of brine

Question

Two very large tanks A and B are each partially filled with 200 gallons of brine. Initially, 25 pounds of salt are dissolved in Tank A while 75 pounds are dissolved in Tank B. One interconnecting pipe sends brine from Tank A to Tank B at a rate of 5 gal/min. Another interconnecting pipe sends brine from Tank B to Tank A at a rate of 2 gal/min. An input pipe feeds pure water to Tank A at a rate of 3 gal/min, while an output pipe drains brine from Tank B at a rate of 3 gal/min. I need to make a mathematical model for the amount of salt X1 and Xb at time in Tanks A and B

Explanation / Answer

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Suppose flow is at the rate of x gallons/min from tank A to tank B.

If x1 is the salt at any time t in tank A, we will have dx1/dt = -xt as salt concentration is 1lbs/gallon

Similarly for tank B, we will have dx2/dt =xt.

Initial conditions are x1 and x2 at t = 0 and x1+x2 = 150 at any time is known

If flow rate is known, we can solve these equations and find x1 and x2 as function of time.

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