At the beginning of this lesson, it's nice to have some sort of creative opener. You might spur student interest with something like the pain relief image here* which sparks students' wondering.
I like to act like I actually have a headache to open up a dialogue along these lines:
"Today, I have a headache. Should I take something for it? If I do, how long will it take for the medicine to kick in? If my headache comes back, how long do I have to wait until I can take another? Does it really matter if I follow the directions on the pill bottle? (YES!) But who comes up with those directions and how?"
All of my students can relate to this situation, but they never would have guessed that a rational function can be used to model it!
* 101questions is a website designed for the purpose of engaging students in problem solving!
Hand out Pharmacokinetics. (By the way, pharmacokinetics is a fancy word for what your body does to a drug, as opposed to pharmacodynamics, which is what a drug does to your body. In this case, we're looking at how the drug passes into and out of the bloodstream.)
Beginning individually or in groups of two or three, students will begin by considering the given function from a mathematical point of view (i.e. graph, asymptotes, etc.). Then they will begin to draw conclusions from the graph about the scenario, examining the maximum value, and solving the equation to find precise times for various concentrations. (MP 4, MP 2)
The emphasis today is not on creating the equation (that is found through clinical trials with the drug), but on using it to draw conclusions about dosage and effective time for the drug. Be sure to point out that while students are expected to find exact times algebraically in parts (c) and (e), they are not expected to solve part (d) algebraically. Students should make use of whatever tools are appropriate in order to estimate the actual maximum. This may include using graphing calculators, an online tool like GeoGebra, or simply evaluating the function at various t-values. (MP 5) In any case, the most important thing is that students are able to correctly interpret the highest point on the graph in context.
Projecting a graph of this function on the board for all to see, I will use the final ten minutes to discuss the solutions to the problem with the class. I'll call on individual students (or groups) to provide answers & explanations, and then ask the rest of the class whether they agree. For the most part, I don't expect many disagreements.
The most likely difficulties are with using the quadratic formula to solve for t in parts (c) and (d). I also expect some students to mistakenly give the earlier time as the answer to part (d) rather than the later time.
Most students will have found a good approximation for the maximum concentration, and this is a good time to discuss and compare various methods for finding this approximation (including ones that don't involve technology).
During this discussion, it's also a good idea to ask students if they can identify the limitations of this model.
A good extension question for the class is to ask what the graph would look like if another dose were taken at the earliest possible moment? Be sure that they see that the maximum concentration would be higher for the second dose than for the first.