## 7.2.1 Modeling the Situation

An interactive environment is used to become familiar with the parameters involved and the range of results that can be obtained.

The interactive figure calculates the amount of medicine in a person's body immediately after taking a dose. In this scenario, the individual takes an initial dose of medicine followed by recurring doses, taken faithfully at fixed intervals of time. The interactive figure allows the initial dose to be different from the recurring doses.

The simulation requires three inputs:

- The initial dose—the amount of medicine given for the initial dose
- The elimination rate—the percent of medicine (given as a decimal) that the kidneys remove from the system between doses
- The recurring dose—the amount of the medicine to be given at fixed intervals

Click Calculate.

*A*(*n*) represents
the amount of medicine in the body immediately *after* taking the *n*th
recurring dose of medicine. So *A*(0) = initial dose, *A*(1) = amount
of medicine in body after first recurring dose, *A*(2) = amount
of medicine in body after second recurring dose, and so on.

This interactive figure calculates
the amount of drug in the system just *after* taking a dose of medicine.
If the amount of medicine in the body just *before* taking the *n*th
dose is desired, subtract the amount of the recurring dose (or initial dose
for *n* = 0) from the value calculated for *A*(*n*).

**Tasks**

- A student strained her knee in an intramural volleyball game, and
her doctor has prescribed an anti-inflammatory drug to reduce the
swelling. She is to take two 220-mg tablets every 8 hours for 10 days.
Her kidneys eliminate 60% of this drug from her body every 8 hours.
Assume she faithfully takes the correct dosage at the prescribed regular
intervals. The interactive figure below contains the initial dose
(440), the elimination rate (0.60), and the recurring dose (440). Click
on Calculate to generate values for the amount of medicine in her body
just after taking each dose of medicine

The interactive figure calculates the amount of drug in the system just after taking a dose of medicine. You could also ask how much drug is in the body just before taking each dose. These values would be exactly 440 mg less than the values calculated just after taking each dose.- How much of the drug is in her system after 10 days, just after she takes her last dose of medicine? If she continued to take the drug for a year, how much of the drug would be in her system just after she took her last dose?
- Does the amount of medicine in the body change faster around the fifth interval (about 40 hours after the initial dose) or around the twenty-fifth interval? How can you tell? What happens to the change in the amount of medicine in the body as time progresses?
- Explain, in mathematical terms and in terms of body metabolism, why the long-term amount of medicine in the body is reasonable.

- Vary the initial dose, the elimination rate, and the recurring dose. What do you notice?

**Discussion**

The interactive figure in this example illustrates
calculation features that can be implemented in spreadsheets or graphing
calculators. Spreadsheets or calculators with iterative capabilities
can be very useful for investigating and understanding change—whether it
is due to growth or to decay. In computer and calculator spreadsheet
programs, students have a powerful tool that permits them to calculate
the results of multiple dynamic events quickly and accurately. The ease
of calculation frees students to focus on the effect of changing one or
more of the problem parameters. In this example, an athlete takes a
constant dose of medicine at regular intervals. Using a calculator or a
spreadsheet, students can determine the effect when changes are made in
the initial dose, the recurring dose, or the percent of medicine
eliminated from the body.

Obtaining explicit formulas that
capture such effects is often quite difficult and in some cases,
impossible. In order to have had the experience that will lead them to
an appropriate closed-form equation with which to model such situations,
students generally must be at a fairly high level of mathematics. A
recursive approach, especially when supported by a calculator or an
electronic spreadsheet, gives students access to interesting problems
such as this earlier in their schooling. It also informally introduces
them to an important mathematical concept—limit.

In this initial
phase of the investigation, students should recognize that the level of
medicine in the body initially rises rapidly but with time increases
less rapidly. Although one might question whether the accuracy of the
recorded answer affects this observation, it appears that the level
eventually stabilizes, so that after about seventeen periods, the value
seems no longer to change. In other words, the athlete's body is
eliminating the same amount of medication as she is taking. This
observation can be mathematically verified by showing that (733 1/3)0.6 =
440.

As students play with the various parameters in the
problem, they might make a number of observations. For example, the
initial dose has no long-term effect; the level of medication still
stabilizes at the same value, regardless of the value of the initial
dose. Changing the recurring dose does change the level at which it
stabilizes. This line of discussion is extended in the next part of the
example.

**Take Time to Reflect**

- What are the advantages and disadvantages of defining the relationship recursively? How might a recursive definition link with other experiences that students have had?
- What particular problems does the concept of limit pose for students? How might this context help them begin to approach this important topic?
- In what ways does technology enhance this investigation? In what ways does it detract from it?

**Reference**

National Research Council.

High School Mathematics at Work: Essays and Examples for the Education of All Students. Washington, D.C.: National Academy Press, 1998.