Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully a

Question

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|. Suppose that the half-life of hydrocodone bitartrate in the body is 3.9 hours, and that the oral does taken is 12 mg. 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. (Write your answer in terms of Q ad t e.g,. Q' = 3t (1-kQ).) Solve your differential equation, assuming that at t = 0| the patient has just absorbed the full 12 mg dose of the drug. (Your solution may include the constant of proporationality k. but should not include any other unspecified constant.) Use the half-life to find the cosntant of proportionality k. How much of the 12 mg does is still in the body after 12 hours? (include units)

Explanation / Answer

a)Q' =-kQ

b)dQ/dt =-kQ

dQ/Q =-k dt

integrate

lnQ=-kt +c

Q=e-kt+c

Q=Ce-kt

Q(0)=12mg

Q=12e-kt

c)half life =3.9hrs

12/2=12e-k*3.9

e-k*3.9=1/2

-3.9k=ln(1/2)

k=0.17773

d)Q=12e-0.17773t

after 12 hours

Q=12e-0.17773*12

Q=1.422mg

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