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 (mg)
0;1000
5;550
10;316
15;180
20;85
25;56
30;31

Answer 1

The data follow an exponential decay A(t) = A0 e^{-k t}. Take natural logs: ln A = ln A0 − k t and fit a line to (t, ln A). A least-squares fit gives

ln A ≈ 6.8957 − 0.11623 t,

so k ≈ 0.11623 per hour. Thus the medication decays at about 11.6% per hour (continuous rate). The fitted model is A(t) ≈ 988.6 e^{-0.11623 t}, and the half-life is t1/2 = ln 2 / k ≈ 5.96 hours.