Question

Q7 After a person takes medicine, the amount of drug left in the person’s body changes over time. When testing a new drug, a pharmaceutical company develops a mathematical model to quantify this relationship. To find such a model, suppose a dose of 1000 mg of a certain drug is absorbed by a person’s bloodstream. Blood samples are taken every five hours, and the amount of drug remaining in the body is calculated. The data collected from a particular sample is recorded below. What is the rate of decay of the medication?
hours since drug was taken, amount left in body
0,1000
5,550
10,316
15,180
20,85
25,56
30,31

Answer 1

Assume exponential decay A(t) = 1000 e^{-k t}. Taking natural logs and doing a least-squares fit of ln(A) vs. t gives slope −k ≈ −0.11643 hr⁻¹.

So k ≈ 0.1164 per hour (≈11.64% per hour), and the fitted model is
A(t) ≈ 1000 e^{-0.1164 t}.

The corresponding half-life is t1/2 = ln 2 / k ≈ 0.6931 / 0.1164 ≈ 5.95 hours.