Question

You have an ear intection and are told to take a \(250 \mathrm{mg}\) tablet of ampicillin (a common antibiotic) four times a day (every six hours). It is known that at the end of six hours, about \(1.6 \%\) of the drug is still in the body. \({ }^{8}\) Let \(Q_{n}\) be the quantity, in milligrams, of ampicillin in the body right after the \(n^{\text {th }}\) tablet. Find \(Q_{3}\) and \(Q_{40}\).

Short answer

Answer: The amount of ampicillin in the body right after taking the third tablet is approximately 254.064 mg, and the amount right after taking the 40th tablet is approximately 253.292 mg.

Step-by-step solution

Set up the formula for the total amount of ampicillin in the body after the n-th tablet

We know that the quantity of ampicillin in the body right after taking a tablet is the sum of the amount of drug added and the amount left from previous tablets. Since \(1.6 \%\) of the drug is left after six hours, after taking the first tablet, \(98.4 \%\) of the drug is eliminated from the body. For the first tablet, the total amount is \(250\ \mathrm{mg}\) (as no drug is taken before). Before taking the second tablet, \(98.4 \%\) of the first tablet is eliminated, so \(1.6 \%\) of \(250\ \mathrm{mg}\) is left. After taking the second tablet, the total amount is the sum of the new \(250\ \mathrm{mg}\) and the remaining amount of the first tablet. In general, after taking the \(n^{\text{th}}\) tablet, the amount of ampicillin in the body just after taking it, denoted as \(Q_n\), can be expressed as follows: $$Q_n = 250 \sum_{i=0}^{n-1}(0.016)^i$$

Calculate \(Q_3\)

Now we substitute \(n = 3\) into our formula to find \(Q_3\): $$Q_3 = 250 \sum_{i=0}^{2}(0.016)^i = 250(1 + 0.016 + (0.016)^2)$$ Now, we can calculate the value: $$Q_3 = 250(1 + 0.016 + 0.000256) = 250(1.016256) \approx 254.064\,\mathrm{mg}$$

Calculate \(Q_{40}\)

Now we substitute \(n = 40\) into our formula to find \(Q_{40}\): $$Q_{40} = 250 \sum_{i=0}^{39}(0.016)^i$$ Using the sum of geometric series formula, \(\frac{1-(r)^n}{1-r}\), where \(r = 0.016\) and \(n = 40\): $$Q_{40} = 250\frac{1-(0.016)^{40}}{1-0.016}$$ Now, we can calculate the value: $$Q_{40} \approx 250\frac{1-0.016^{40}}{0.984} = 253.292\,\mathrm{mg}$$

Present the results

The amount of ampicillin in the body right after taking the third tablet, \(Q_3\), is approximately \(254.064\,\mathrm{mg}\), and the amount right after taking the 40th tablet, \(Q_{40}\), is approximately \(253.292\,\mathrm{mg}\).

Key concepts

Exponential Decay

Exponential decay is a concept that describes how a quantity decreases rapidly over time at a rate proportional to its current value. In the context of medication, such as the antibiotic ampicillin, exponential decay can be observed in the way the drug concentration in the body reduces over time. After taking a dose, the body metabolizes and eliminates the drug, causing its concentration to decrease exponentially.

In our exercise, it's noted that only 1.6% of the ampicillin remains in the body after six hours. This indicates that the drug decays exponentially, with 98.4% being eliminated in each six-hour cycle. The decay can be represented using a factor, often denoted as the decay constant. For ampicillin, this constant can be calculated based on the percentage retained, which here is 0.016 (since 1.6% equals 0.016 in decimal).

This understanding allows us to set up models in medicine to predict how much of a drug remains in the body at any given time, which is key for effective dosing and treatment plans.

Drug Dosage Calculations

Calculating drug dosages accurately is critical for ensuring the safety and effectiveness of a treatment. In the context of the ampicillin exercise, understanding how much of the drug remains in the body after multiple doses is a part of dosage calculations.

For instance, after the first 250 mg tablet, there is no drug previously in your system. Before the next dose, only 1.6% of the first dose remains. By adding another 250 mg with each dose, we need to calculate the cumulative amount in the body.

• Initial Dose: 250 mg.
• Elimination: 98.4% per 6 hours, meaning each subsequent dose retains more from prior doses.

The formula to calculate the dosing pattern and the amount like in our exercise, often involves adding the current dose to the amount left from previous doses. This ensures sustained therapeutic levels of the drug in the body without reaching toxic levels.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the drug dosage exercise, we utilize a geometric series to determine the total amount of ampicillin remaining in the body.

The geometric series in this context is formed by each amount of drug remaining in the body after a dose at intervals, with a ratio of 0.016 (1.6%).

• First term: full dose of 250 mg.
• Common ratio: 0.016, indicating how much remains after each period.

When calculating quantities like \(Q_3\) or \(Q_{40}\), the formula for the sum of a finite geometric series is highly useful. This formula allows for easy calculation of total amounts by summing up all the remaining parts of each dose over time, offering an efficient way to handle multiple doses in a comprehensive treatment plan.