First, interpret the formula (the model is exponential): A(t) = 0.325(0.76)t means the amount of acetaminophen A(t) (in grams) after t hours is the initial 0.325 g multiplied by 0.76 raised to the t power.
Step-by-step meaning:
- The factor 0.76 is the remaining fraction after one hour. So each hour the body keeps 76% of the previous hour's amount and eliminates 24% of it.
- Because the factor is raised to the t power, the drug amount decreases multiplicatively each hour — this is exponential decay.
Examples:
- After 1 hour: A(1) = 0.325 × 0.76 = 0.247 g (about 76% of the original).
- After 2 hours: A(2) = 0.325 × 0.762 = 0.325 × 0.5776 ≈ 0.1877 g.
- After 3 hours: A(3) = 0.325 × 0.763 ≈ 0.325 × 0.438976 ≈ 0.1427 g.
Half-life (time to reach half the initial amount): solve 0.76t = 0.5 so t = ln(0.5) / ln(0.76) ≈ 2.525 hours. That means roughly every 2.53 hours the acetaminophen amount falls to half its previous value.
Long-term behavior: as t → ∞, 0.76t → 0, so A(t) → 0. In other words, the drug amount keeps decreasing and approaches zero over time.
Summary: The amount of acetaminophen decays exponentially — 76% remains each hour (a 24% reduction per hour), with a half-life of about 2.53 hours, and the amount approaches zero as time passes.