You have an ear infection and are told to take a 200 mg tablet of ampicillin (a common antibiotic) four times a day (every six hours). It is known that at the end of six hours, about 5% of the drug is still in the body. What quantity of the drug is in the body right after the the 3rd tablet?The 12th Tablet?
Assuming you continue taking tablets, what happens to the drug level in the long run?
Round your answers to three decimal places.
____ mg after the 3rd tablet
____ mg after the 12th tablet
____ mg in the long run
To find the quantity of the drug in the body after the 3rd tablet, we need to calculate the amount of drug remaining in the body after each tablet is taken.
The amount of drug remaining after each tablet can be calculated using the formula:
Remaining amount = Initial amount * (1 - Percentage remaining)^Number of tablets
In this case, the initial amount is 200 mg, and the percentage remaining is 0.05 (since 5% of the drug remains in the body after 6 hours).
1) After the 1st tablet:
Remaining amount = 200 * (1 - 0.05)^1 = 200 * 0.95 = 190 mg
2) After the 2nd tablet:
Remaining amount = 190 * (1 - 0.05)^1 = 190 * 0.95 = 180.5 mg
3) After the 3rd tablet:
Remaining amount = 180.5 * (1 - 0.05)^1 = 180.5 * 0.95 = 171.475 mg
Therefore, after the 3rd tablet, there would be 171.475 mg of the drug remaining in the body. (Round to three decimal places: 171.475 mg)
To find the quantity of the drug in the body after the 12th tablet, we can continue using the same formula.
4) After the 4th tablet:
Remaining amount = 171.475 * (1 - 0.05)^1 = 171.475 * 0.95 = 162.90125 mg
5) After the 5th tablet:
Remaining amount = 162.90125 * (1 - 0.05)^1 = 162.90125 * 0.95 = 154.7561875 mg
.... and so on until we reach the 12th tablet.
12) After the 12th tablet:
Remaining amount = 131.019 * (1 - 0.05)^1 = 131.019 * 0.95 = 124.46805 mg
Therefore, after the 12th tablet, there would be 124.46805 mg of the drug remaining in the body. (Round to three decimal places: 124.468 mg)
In the long run, the drug level will eventually reach a steady-state where the amount eliminated through each dose is equal to the amount absorbed by each dose. This means that the amount of drug in the body will stabilize and remain relatively constant over time.
The steady-state drug level can be approximated by dividing the dosage by the elimination rate:
Steady-state drug level = Dosage / Elimination rate
= 200 mg / 0.05
= 4000 mg
Therefore, in the long run, the drug level will reach approximately 4000 mg. (Round to three decimal places: 4000 mg)
To determine the quantity of the drug in the body after the 3rd and 12th tablet, we need to consider the percentage of the drug that remains in the body after each six-hour period.
After each six hours, 5% of the drug remains in the body. Therefore, after the first six-hour period (following the first tablet), the quantity of the drug remaining in the body is 95% of the initial dose.
To calculate the quantity of the drug after the 3rd tablet:
1. Start with the initial dose of 200 mg.
2. After the first tablet, 95% of the drug remains: 200 mg * 0.95 = 190 mg.
3. After the second tablet, 190 mg * 0.95 = 180.5 mg.
4. After the third tablet, 180.5 mg * 0.95 = 171.48 mg.
Therefore, after the 3rd tablet, approximately 171.48 mg of the drug is in the body.
To calculate the quantity of the drug after the 12th tablet:
1. Start with the initial dose of 200 mg.
2. After each tablet, 95% of the drug remains.
3. After the 12th tablet, the quantity of the drug can be calculated as: 200 mg * (0.95)^12 ≈ 102.784 mg.
Therefore, after the 12th tablet, approximately 102.784 mg of the drug is in the body.
Now, let's consider what happens to the drug level in the long run. Since 5% of the drug remains in the body after each six-hour period, the quantity of the drug in the long run will eventually stabilize at a certain level.
To find this steady-state level, we can set up an equation:
X = 0.05X + 200
where X represents the steady-state drug level.
Simplifying the equation:
X - 0.05X = 200
0.95X = 200
X ≈ 210.53 mg
Therefore, in the long run, the drug level will stabilize at approximately 210.53 mg.
In summary:
- After the 3rd tablet: Approximately 171.48 mg of the drug is in the body.
- After the 12th tablet: Approximately 102.784 mg of the drug is in the body.
- In the long run: The drug level stabilizes at approximately 210.53 mg.