Codeine phosphate is a drug used as a painkiller. A common brand contains 30 mg of codeine. 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): 30, 60, 90, 120, 150, 180
Amount of Codeine in Blood (mg): 27.0, 23.5, 21.2, 18.7, 16.6, 14.5
I figured out the answers for parts a, b and c, but I need help with part d and e.
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 body.
e) If the patient were to delay taking the second pill, how would it affect the results from part d)?
The 2nd pill is shifted by 60 minutes.
So, if the 1st pill is modeled by f(x), the 2nd pill will be modeled by f(x-t).
Now just work with the new f(x).
To answer part d and e of the given question, let's first recap the information provided:
Time After Consumption (min): 30, 60, 90, 120, 150, 180
Amount of Codeine in Blood (mg): 27.0, 23.5, 21.2, 18.7, 16.6, 14.5
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.
To create a model for the amount of codeine in the patient's bloodstream t minutes after taking the first pill, we can continue to use the same equation derived in part a.
From the scatter plot or calculation, determine the equation that best fits the data points. Let's assume we have determined the equation as follows:
Amount of Codeine = a * t^n + c
Based on the given data, we can substitute the values of time (t) and amount of codeine for the first pill to determine the values of a, n, and c in the equation. Once we have these values, we can use the same equation to find the amount of codeine for any given time after taking the first pill.
To determine the maximum amount of codeine in the patient's bloodstream, substitute the time at which the maximum amount occurs into the equation. The resulting amount of codeine will give you the maximum value.
To determine when 90% of the maximum amount would be eliminated from the body, multiply the maximum amount of codeine by 0.9 and solve the equation for t.
e) If the patient were to delay taking the second pill, how would it affect the results from part d)?
If the patient delays taking the second pill, it would impact the overall amount of codeine in the bloodstream. The time at which the maximum amount of codeine occurs and the time at which 90% of the maximum amount is eliminated from the body would both be affected. The delay in taking the second pill would cause a shift in the timing of these events.
To determine the exact impact on the results from part d, you would need to recalculate the equation using the new time when the second pill is taken. This would require updating the values of a, n, and c in the equation based on the new data.
Remember that the model assumes the same rate of codeine elimination applies to both pills. If there are factors that may affect codeine metabolism or absorption, these should be considered in analyzing the results.