Allergy medication

Task

Every day Brian takes 20 mg of a drug that helps with his allergies. His doctor tells him that each hour the amount of drug in his blood stream decreases by 15%. 

1. Construct an exponential function of the form $f(t) = ab^t$, for constants $a$ and $b$, that gives the quantity of the drug, in milligrams, that remains in his bloodstream $t$ hours after he takes the medication.
2. How much of the drug remains one day after taking it?
3. Do you expect the percentage of the dose that leaves the blood stream in the first half hour to be more than or less than 15%? What percentage is it?
4. How much of the drug remains one minute after taking it?

IM Commentary

When students first start studying exponential functions they evaluate them at integer inputs. After they have learned about rational exponents they can evaluate them at other inputs. The purpose of the task is to help students become accustomed to evaluating exponential functions at non-integer inputs and interpreting the values. They do this by constructing a model to interpret a real-world situation with an emphasis on evaluating at non-integer values.

There are several opportunities to attend to precision MP.6 in this task. First of all, all of the outputs of the function used to model the amount of the drug left in the patient's bloodstream are approximations. (How quickly a body processes chemicals can be affected by a range of factors, including the amount of exercise it gets and the presence of other chemicals in its system.) This is indicated by the presence of $\approx$'s in the solution, but the issue might bear a few moments of attention. Secondly, the task does not include rounding instructions and calculated answers aren't "nice" numbers, so the teacher can decide whether to prescribe that students round to the tenths place, as shown in the solution, or decide for themselves what an appropriate level of precision would be. 

Although part (c) doesn't elicit this observation explicitly, a nice thing to notice is that the percentage of the drug that leaves the bloodstream in the first half an hour, 8%, is not half of 15%, because the decrease won't behave "linearly" each hour -- the drug leaves faster at first, so more than half of 15% leaves in the first half-hour. A teacher could opt to prompt students to explain why more of the drug leaves in the first half-hour than the second half-hour. A teacher could push students further to think in terms of exponential functions decreasing by equal factors over equal intervals, so if the factor for an hour is 0.85, the factor for a half hour is $\sqrt{0.85}$.

Solution

1. Since the amount of the drug is 20 mg when $t=0$, we have $f(0) = ab^0 = a \cdot 1 = a = 20$. Each hour the quantity of drug decreases by 15%, which is the same thing as being multiplied by $1-0.15 = 0.85$, so the decay factor is $0.85$ and we have $f(t) = 20(0.85)^t$.
2. After 24 hours the amount remaining is $f(24) = 20(0.85)^{24} \approx 0.4$ mg.
3. We would expect the percentage that leaves the blood stream in the first half hour to be less than the 15% which leaves in the first hour. Half an hour corresponds to $t = 1/2$, so after half an hour the amount remaining is $f(1/2) = 20 (0.85)^{1/2} \approx 18.4$ mg. This is $18.4/20 = 92\%$ of the original dose, so approximately 8% has left the blood stream.
4. One minute is $1/60$ of an hour, so the amount remaining after one minute is $f(1/60) = 20 (0.85)^{1/60} \approx 19.9$ mg, very close to the original dose as we would expect.