4. (45 points) Two large tanks A and B are interconnected. Initially, tank A contains 24...

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4. (45 points) Two large tanks A and B are interconnected. Initially, tank A contains 24 liters of brine which contains 5 kg of dissolved salt and tank B contains 24 liters of brine in which 2 kg of salt is dissolved. Starting at time t= 0, • pure water flows into tank A at the rate of 6 liters/min. • brine flows from tank A into tank B at the rate of 8 liters/min. • brine is pumped from tank B back into tank A at the rate of 2 liters/min. . brine flows out of tank B and away from the system at the rate of 6 liters/min. The mixture in the tanks is kept uniform by stirring. Let r (t) denote the amount of salt in tank A at any time t>0 and let y (t) denote the amount of salt in tankB at any time t>0. (i) Draw a nice large diagram of the interconnected tanks. (ii) Formulate an IVP that models the problem. (iii) Use the Laplace Transform Method to solve the IVP to determine the amount of salt in each tank at any time t>0. Express your answer in the form z (t) = Aeat + Be y (t) = Ceat + Deb, %3D where A, B, C, D, a, and b are constants. (iv) For t > 0, sketch large graphs of z (t) and y (t) on the same axes and determine the amount of salt in each tank after a very long time. (v) After how many minutes do the two tanks contain the same amount of salt? Determine this amount of salt. (vi) Which tank contains more salt after 24 min? Does this surprise you? Explain. 4. (45 points) Two large tanks A and B are interconnected. Initially, tank A contains 24 liters of brine which contains 5 kg of dissolved salt and tank B contains 24 liters of brine in which 2 kg of salt is dissolved. Starting at time t= 0, • pure water flows into tank A at the rate of 6 liters/min. • brine flows from tank A into tank B at the rate of 8 liters/min. • brine is pumped from tank B back into tank A at the rate of 2 liters/min. . brine flows out of tank B and away from the system at the rate of 6 liters/min. The mixture in the tanks is kept uniform by stirring. Let r (t) denote the amount of salt in tank A at any time t>0 and let y (t) denote the amount of salt in tankB at any time t>0. (i) Draw a nice large diagram of the interconnected tanks. (ii) Formulate an IVP that models the problem. (iii) Use the Laplace Transform Method to solve the IVP to determine the amount of salt in each tank at any time t>0. Express your answer in the form z (t) = Aeat + Be y (t) = Ceat + Deb, %3D where A, B, C, D, a, and b are constants. (iv) For t > 0, sketch large graphs of z (t) and y (t) on the same axes and determine the amount of salt in each tank after a very long time. (v) After how many minutes do the two tanks contain the same amount of salt? Determine this amount of salt. (vi) Which tank contains more salt after 24 min? Does this surprise you? Explain.

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