Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body. The half-life of hydrocodone bitartrate in the body is 3.8 hours and the dose is 10 mg.
(a) Write a differential equation for the quantity, Q, of hydrocodone bitartrate in the body at time t, in hours since the drug was fully absorbed.
(b) Solve the differential equation given in part (a).
(c) Use the half-life to find the constant of proportionality, k.
(d) How much of the 10-mg dose is still in the body after 12 hours?