{Use of Tech} Periodic dosing
Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.

b.Use a calculator to estimate this limit. In the long run, how much drug is in the person’s blood?

검증된 단계별 안내

Identify the problem as a periodic dosing model where a fixed amount of aspirin (80 mg) is taken every 24 hours, and the drug decays by half every 24 hours due to its half-life.

Express the amount of aspirin in the blood right after each dose as a sequence \( A_n \), where \( A_n \) is the amount immediately after the \( n^{th} \) dose.

Set up the recursive relation for the sequence: after 24 hours, half of the previous amount remains, and then 80 mg is added. This gives \( A_{n} = \frac{1}{2} A_{n-1} + 80 \).

Recognize that this is a linear difference equation and find its long-term behavior by solving for the steady-state (limit) \( L \) where \( L = \frac{1}{2} L + 80 \).

Solve the equation for \( L \) to find the equilibrium amount of aspirin in the blood after many doses, which represents the long-run amount of drug present.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Decay and Half-Life

Half-life is the time required for a quantity to reduce to half its initial value, modeling exponential decay. In this problem, aspirin’s half-life of 24 hours means the drug amount halves every day, which can be expressed using exponential decay formulas to track the drug concentration over time.

Geometric Series and Limits

Repeated dosing with decay leads to a sum of decreasing amounts forming a geometric series. Understanding how to sum an infinite geometric series and find its limit is essential to determine the steady-state amount of drug in the blood after many doses.

Steady-State Concentration in Pharmacokinetics

Steady-state concentration occurs when the amount of drug administered equals the amount eliminated over a dosing interval. Calculating this equilibrium helps estimate the long-term drug level in the bloodstream, crucial for understanding the drug’s effectiveness and safety.