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Differential Equation: Laplace Translation Theorems

1. (12) Three tanks are filled with a dye solution. Tanks $A$ and B initially contains $20hg$ of dye and tank $C$ is initially pure water. A solution that contains $3hg/L$ of dye is pumped into tank $A$ at a rate of $50L/hr$. The solution in tank $A$ flows out to tank $B$ at a rate of 40 $L/hr$ and out an exhaust spout at a rate of $20L/hr$. The solution in tank B flows into tank A at a rate of $10L/hr$, into tank $C$ at a rate of $30L/hr$, and out an exhaust spout at a rate of $10L/hr$. The solution in tank $C$ flows into $tankB$ at a rate of $10L/hr$ and out an exhaust spout at a rate of $20L/hr$. Tank $A$ holds $200L$, tank $B$ holds $100L$, and tank $C$ holds $100L$. Set up a system to determine the amount of dye in each tank at a given time $t$. Write your final system in matrix form. "You do not need to solve the system.

Given data:

Three tanks are filled with a dye solution. Tanks and initially contains hg of dye and tank is initially pure water. A solution that contains of dye is pumped into tank at a rate of The solution in tank flows out to tank at a rate of and out an exhaust spout at a rate of . The solution in tank flows into tank at a rate of , into tank at a rate of , and out an exhaust spout at a rate of

To find:

Set up a system to determine the amount of dye in each tank at a given time .

Write your final system in matrix form.

Now,

Let denote the amounts (in hg) of dye in tanks and , respectively.

Both and start with of dye, so .

starts off containing only pure water, so