5. (20 points) Recall from the single tank problems: = (rate in x concentration in) - (rate out x concentration out) Use this idea to expand to a multiple tank system. Tank D initially contains 800 liters of liquid that is 65% toxin and the rest water. Tank P initially contains 300 liters of pure water. 20 liters of liquid per minute with concentration 5 grams of toxin per liter is pumped into tank D from the Los Angeles River. At the same time 20 liters of mixture is pumped out of tank P into the Back Bay (not into any tank). There are two pipes that connect tank D and tank P. One pipe pumps mixture from tank D to tank P at a rate of 35 liters per minute. The other pipes pumps from tank P to D at a rate of 15 liters per minute. Notice the amount of liquid stays constant in each of the tanks. a. Let x(t) be the amount of toxin (grams) in tank D at time t. Write, this could be a function of any of the variables: x, y, and/or t. b. Let y(t) be the amount of toxin (grams) in tank P at time t. Write, this could be a function of any of the variables: x, y, and/or t. c. Write these equations as a first order differential matrix equation. What are the initial conditions for r(t) and y(t)? d. Consider the system of equations, is there a fixed point of the system? Justify. If so, what is the fixed point? Verbally interpret what a fixed point was represent in the context of this problem.

Fig: 1