Mathematics 132 – Calculus for Physical and Life Science 2 Group Work on Differential Equation Models March 19, 2008

The Story Homer needs an operation to clear carotid arteries blocked by fatty plaque deposits

(these are the arteries in the neck that feed blood to the brain, so it’s pretty serious!). When he is brought into the hospital, he is given a sedative to help him sleep. The doctor wants to operate as soon as possible, but cannot safely do so until the concentration of sedative in Homer’s body is less than 0.03 milligrams per liter. You will use the following table of data to determine how many hours the doctor will need to wait to start the operation. Blood samples were taken every 10 minutes and the sedative concentration was determined in milligrams per liter:

time (minutes) Concentration (mg/L)
0  10.0
10  6.07
20  3.68
30  2.23
40  1.35
50  .820
60  .498
70  .302
80  .183
90  .111

Questions

1. Is the rate of change of the concentration of the sedative a constant? Explain.

2. Estimate the rate of change of the concentration of the sedative at 30 minutes, 60 minutes, and 80 minutes.

3. Show that the rate of change of the concentration of the sedative is (roughly) proportional to the concentration at the times from question 2.

4. Assuming that the pattern remains true for all times, write the relationship given in question 3 as a differential equation for the function C(t) = concentration of the sedative at time t.

5. What is numerical value of the constant of proportionality in your equation?

6. Solve the differential equation from question 4 using the constant from question 5.

7. When is the concentration of sedative half of what it was at t = 0?

8. If Homer received the sedative at 2:00 pm, when can the doctor start the operation?

Turn in your answers at the end of the period.