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.