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May 29th, 2018, 08:08 PM   #1
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Challenging Derivatives of Exponential and Sinusoidal Functions task question

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 attached.

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)?

Last edited by skipjack; May 30th, 2018 at 12:04 AM.
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May 29th, 2018, 11:55 PM   #2
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I'd definitely use Zorn's lemma.
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May 30th, 2018, 01:14 PM   #3
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Try exponential first $30e^{-kt}$.
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May 31st, 2018, 04:18 AM   #4
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That was my thinking as well. Curiously, $30.4e^{-kt}$ provides a better fit to the given data.
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