Compute the maximum and minimum amount of a drug remaining in a patient
in the limit (that is, as time tends to infinity) given that the patient takes an 80 milligram dose
once a day at the same time each day and the drug has a half life of 22 hours.
starting at time 0, when the first dose is taken, the amount after t hours is
80*(1/2)^(t/22)
the 2nd dose:
80*(1/2)^((t-24)/22)
the nth dose:
80*(1/2)^((t-24n)/22)
Play around with that as t gets large.
To compute the maximum and minimum amount of the drug remaining in a patient in the limit (as time tends to infinity), we can use the concept of half-life.
The half-life of a drug is the amount of time it takes for half of the drug to be eliminated from the body. In this case, the drug has a half-life of 22 hours, meaning that after 22 hours, half of the drug dose will be eliminated from the body.
To calculate the maximum amount of the drug remaining in the patient, we assume that the drug is fully absorbed and distributed throughout the body. Therefore, after each dose, the amount of the drug remaining in the patient will be equal to the dose initially taken.
To calculate the minimum amount of the drug remaining in the patient, we consider that after each dose, half of the drug is eliminated. So, after each 22-hour period, half of the drug remaining from the previous dose is eliminated.
Let's calculate:
Maximum amount of the drug remaining:
Since the drug is fully absorbed and distributed, the maximum amount of the drug remaining in the patient is equal to the initial dose, which is 80 milligrams.
Minimum amount of the drug remaining:
After each 22-hour period, half of the drug remaining is eliminated. Therefore, each day, the amount of drug remaining will be:
Day 1: 80 mg remaining
Day 2: 40 mg remaining
Day 3: 20 mg remaining
Day 4: 10 mg remaining
and so on...
As time tends to infinity, the minimum amount of the drug remaining will approach zero but will never actually reach zero.
So, in the limit (as time tends to infinity):
- The maximum amount of the drug remaining in the patient is 80 milligrams.
- The minimum amount of the drug remaining approaches zero but never reaches zero.
Please note that these calculations assume that the drug is being taken consistently without any changes in dosage or elimination rate. Additionally, it is important to consult a healthcare professional for accurate and personalized information regarding medication and its effects.