## Microsoft word - linear vs. exponential project.docx

Advanced Quantitative Reasoning Coach Whitt Linear vs. Exponential Project
Caffeine Elimination Rate
The human body processes different drugs in different ways. Most of you have probably observed that a drunken person does not stay drunk forever. Sooner or later, the blood alcohol concentration (BAC) returns to zero. Alcohol is a rather unusual chemical in terms of how the body processes it. To understand it better, let’s compare it with the caffeine in an average soft drink. Assume that you quickly drink a 12 oz soft drink containing about 48 mg of caffeine. If the amount of caffeine in your body were monitored over the next 19 hours, the result would be similar to the data in the following table. Time (hrs) Caffeine 48.00 38.14 30.26 24.03 19.08 15.15 12.03 9.55 1. Make a scatter plot of the data (x = time, y = Caffeine) from the table. 2. Examine the scatter plot and the table. How would you best describe the shape of the data? Discuss form, direction, and strength of the correlation. 3. Using your graphing calculator, determine the best-fit regression equation for the graph. Advanced Quantitative Reasoning Coach Whitt 4. Taking note of the quantities you graphed, explain the meaning of the constants a and b in this equation. 5. When will your body have eliminated all the caffeine? Explain how you know.
Caffeine, like most drugs, is classified as a first-order drug because exponential decay accurately models its
elimination from the human body. Alcohol, on the other hand, is a zero-order drug. Let’s look at what the
difference is.
Alcohol Elimination Rate
Assume that 24-year old Bryan has been eating and drinking for most of the evening and by midnight has a blood alcohol concentration (BAC) of 0.19 g/dl (grams/deciliter) – far above the legal driving limit of 0.08 g/dl. If Bryan weighs 140 lbs, it would take approximately seven drinks for his BAC to reach 0.19 g/dl. Fortunately for Bryan, a friend has already taken his car keys. The question now is, when should the keys be given back to Bryan? If someone were to monitor Bryan’s BAC over the next 8 hours, it would most likely look like the data in the following table. Time 6. Make a scatter plot of the data (x = time, y = BAC) from the table. 7. How would you best describe the shape of the data? Discuss form, direction, and strength of the Advanced Quantitative Reasoning Coach Whitt 8. Determine the best-fit regression equation for this graph. 9. Taking note of the quantities you graphed, explain the meaning of the constants a and b in this equation. 10. As mentioned, Bryan wisely stopped drinking at midnight. Legally, Bryan cannot drive home until his BAC is at or below 0.08 g/dl. When will he be sober enough, legally, to drive home? [Note that the 0.08 g/dl BAC level is statutory in all states; however, many states have “zero tolerance” for persons under the age of 21. Under these laws, drivers who are not yet 21 cannot have any alcohol in their body while driving.] 11. When was all the alcohol eliminated from Bryan’s body? How do you know? 12. The elimination rate of alcohol from the human body is highly variable. The rate we found here is an average value, but elimination rates as low as 0.01 g/dl are common. Write a new regression equation using 0.01 as the elimination rate. At this conservative rate, at what time would the alcohol have been eliminated from Bryan’s body? Explain how you determined the answer to this question. Advanced Quantitative Reasoning Coach Whitt 13. What are the differences between zero-order drugs (such as alcohol) and first-order drugs (such as caffeine)? Discuss these differences in terms of how your body handles the drugs, as well as in mathematical terms. 14. Write a short reflection of this investigation, summarizing what you have learned as well as any