Problem 24 - Precalculus

Question

A drug has a half-life of 12 hours. If the initial dosage is 5 milligrams, how many milligrams will be in the patient's body in 16 hours?

Step-by-Step Solution

Answer

Approximately 1.984 milligrams remain after 16 hours.

Step 1: Understand the Concept of Half-life

The half-life of a drug is the time it takes for half of the drug to be eliminated from the body. In this exercise, the half-life is given as 12 hours, meaning every 12 hours, the drug amount reduces to half of its previous amount.

Step 2: Identify Initial Conditions and Time Passed

The initial dosage given is 5 milligrams. We need to determine how many milligrams remain in the body after 16 hours.

Step 3: Calculate the Number of Half-lives Passed

Determine how many half-lives pass in 16 hours. Since one half-life is 12 hours: \[ \text{Number of half-lives} = \frac{16}{12} = \frac{4}{3} \approx 1.33 \]

Step 4: Apply the Half-life Formula

Use the exponential decay formula: \[ A = A_0 \times \left( \frac{1}{2} \right)^n \]where \( A_0 = 5 \) milligrams, \( n = \frac{4}{3} \). Substitute these values into the formula: \[ A = 5 \times \left( \frac{1}{2} \right)^\frac{4}{3} \]

Step 5: Compute the Remaining Dosage

Calculate the amount \( A \) using a calculator:\[ A = 5 \times 0.39685 \approx 1.984 \text{ milligrams} \]Here, the critical computation is that \( \left( \frac{1}{2} \right)^\frac{4}{3} \approx 0.39685 \).

Step 6: Conclude with the Result

After 16 hours, approximately 1.984 milligrams of the drug remain in the patient's body.

Key Concepts

Understanding Half-life

Half-life is a fundamental concept in science, particularly in chemistry and pharmacology. It refers to the time required for a substance to reduce to half its initial amount. In our case, it's about how the concentration of a drug decreases over time.
For example, if a drug's half-life is 12 hours, starting with 5 milligrams means after 12 hours, only 2.5 milligrams will remain in the body. This process continues every 12 hours, cutting the current amount by half each time.

• Half-life is consistent: no matter how much you start with, it always takes the same amount of time to halve the remaining substance.
• It's crucial for understanding how medications process and are eliminated from the body.

Understanding half-life helps predict how long a drug stays active and how often a dosage might need adjusting.

Initial Dosage Explanation

The initial dosage is the starting amount of a drug administered to a patient. It is crucial because it determines the initial concentration of the medicine in the body.
In terms of exponential decay, the initial dosage is denoted as \(A_0\) in the formula. This value is a benchmark that allows us to predict future concentrations after certain time intervals.

• Initial dosage: set by medical professionals based on factors like age, weight, and health condition.
• It's important for effective treatment: too little might be ineffective, too much could be harmful.

Accurate initial dosages ensure that the drug works efficiently while minimizing potential side effects.

Introducing the Exponential Decay Formula

The exponential decay formula plays a crucial role in calculating how a substance reduces over time. It's particularly useful in contexts such as radioactive decay and pharmacokinetics.
The formula is given by:\[A = A_0 \times \left( \frac{1}{2} \right)^n\]Here, \(A\) represents the remaining dosage, \(A_0\) is the initial dosage, and \(n\) is the number of half-lives that have elapsed.

• This formula illustrates how each half-life reduces the amount by 50%.
• It's a reliable tool for predicting decreases in drug concentrations or decay processes.

Using this formula helps ensure precise calculations, crucial for applications like medication management where timing and dosages are critical.