Question 8 (Biological Elimination of Chemical Substances). The way in which the human body breaks down various chemical substances is modeled via exponential decay: over a fixed period of time (called the biological half-life[2]), half of substance is processed and eliminated by the body. For example, caffeine typically has a halflife of about about 5 hours, and a one ounce shot of espresso has about 64 mg of caffeine. Ignoring confounding factors, if a person has a shot of espresso at 6am, their bloodstream will contain around 32 mg of caffeine at noon, around 16 mg of caffeine at 5pm, and so on. (i) Tylenol has a biological half-life of around 2.5 hours. Determine the parameter k such that the amount of Tylenol in a person’s system t hours after taking a dose of the medication is given by A(t)=A02kt . (ii) Suppose that a patient is given 500 mg of Tylenol at noon. How much of the drug will be in their system at 6pm? midnight? noon the next day? (iii) Suppose that Tylenol can no longer be detected in the bloodstream if the amount of the drug falls below a total of 10 mg. If a patient is given 300 mg of Tylenol at 8am, when will it no longer be possible to detect the drug in their system?
W11 Questions 8 and 9
Question 8 (Biological Elimination of Chemical Substances). The way in which the
human body breaks down various chemical substances is modeled via exponential
decay: over a fixed period of time (called the biological half-life[2]), half of substance
is processed and eliminated by the body. For example, caffeine typically has a halflife of about about 5 hours, and a one ounce shot of espresso has about 64 mg of
caffeine. Ignoring confounding factors, if a person has a shot of espresso at 6am,
their bloodstream will contain around 32 mg of caffeine at noon, around 16 mg of
caffeine at 5pm, and so on.
(i) Tylenol has a biological half-life of around 2.5 hours. Determine the parameter k such that the amount of Tylenol in a person’s system t hours after taking
a dose of the medication is given by
A(t)=A02kt .
(ii) Suppose that a patient is given 500 mg of Tylenol at noon. How much of the
drug will be in their system at 6pm? midnight? noon the next day?
(iii) Suppose that Tylenol can no longer be detected in the bloodstream if the
amount of the drug falls below a total of 10 mg. If a patient is given 300 mg
of Tylenol at 8am, when will it no longer be possible to detect the drug in
their system?
Question 9. Continue to work in the context of Question 8: Tylenol has a biological
half-life of about 2.5 hours. The label directions indicate that a patient should take a
400 mg dose every 6 hours. Suppose that a patient takes their first dose at midnight,
and takes another dose every 6 hours, according to these directions. Sketch a graph
of the amount of Tylenol in that patient’s system over the next 48 hours. How much
of the medicine is in the patient’s sytem just before they take a dose at the 48 hour
mark?
Keep in mind that your grammar, spelling, and style are important. Do your scratch
work separately, and write up a “final draft” to turn in.
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