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, with constant of proportionality $k < 0$. Suppose that the half-life of hydrocodone bitartrate in the body is 3.6 hours, and that the oral dose taken 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.
$Q' =$
(Write your answer in terms of $Q$ and $t$, e.g., $Q' = 3 t(1 - k Q)$.)

(b) Solve your differential equation, assuming that at $t=0$ the patient has just absorbed the full 10 mg dose of the drug.
$Q =$
(Your solution may include the constant of proporationality $k$, but should not include any other unspecified constant.)

(c) Use the half-life to find the constant of proportionality, $k$.
$k =$

(d) How much of the 10 mg dose is still in the body after 12 hours?
Amount =
(include units)