Codeine phosphate is a drug used as a painkiller. Generally, it is mixed with acetaminophen in tablet form. It is rapidly absorbed into the bloodstream from the gastrointestinal tract and is gradually eliminated from the body via the kidneys. A common brand contains 30 mg of codeine. Since it is physically addictive and has other unwanted side effects, it is important to avoid an overdose while helping to relieve pain symptoms such as those caused by a headache. Samples of blood were taken at regular time intervals from a patient who had taken a pill containing 30 mg of codeine. The amount of codeine in the bloodstream was determined every 30 min for 3 h. The data are shown in the table below.
| Time After consumption (min) | Amount of Codeine in blood (mg) |
| 30 | 27.0 |
| 60 | 23.5 |
| 90 | 21.2 |
| 120 | 18.7 |
| 150 | 16.6 |
| 180 | 14.5 |
a) Create a scatter plot of the data and determine a suitable equation to model the amount of codeine in the bloodstream t min after taking the pill. Justify your choice of models.
b) Use the model to determine the instantaneous rate of change in the amount of codeine at each time given in the chart. How does it relate to the amount of codeine in the blood?
c) It is recommended that a second pill be taken when 90% of the codeine is eliminated from the body. When would this occur?
d) Assume that the same model applies to the second pill as to the first. Suppose the patient took a second pill one hour after consuming the first pill.
- Create a model for the amount of codeine in the patient’s bloodstream t min after taking the first pill.
- Determine the maximum amount of codeine in the patient’s bloodstream.
- Determine when 90% of the maximum amount would be eliminated from the bod
e) If the patient were to delay taking the second pill, how would it affect the results from part d)? Use Video Note to complete this question, showing all work with a full explanation.
