Question

17. Life Science: drug dosage. A dose of a drug is injected into the body of a patient. The drug amount in the body decreases at a rate of 55% per hour, that is, dA =-0.55A, dt where A is the amount in the body and r is the time, in hours. (a) A dose of 7 cubic centimeters (cc) is administered. Assuming A = 7. find the function that satisfies the equation. (b) How much of the initial dose of 7 cc will remain after 4 hr? (c) After how long does half of the original dose remain? 17. (a) (b) (c)

Answer 1

We have .

dA dt = :-0.55A

a)Rearranding and integrating,

$\frac{\mathrm{d} A}{A}=-0.55\mathrm{d} t\\ \int \frac{\mathrm{d} A}{A}=-0.55\int \mathrm{d} t\\ \ln A=-0.55t+C\\ A=A_0e^{-0.55t}\\ {\color{Blue} A=7e^{-0.55t}}$

b) After 4 hours,

$A(4)=7e^{-0.55(4)}\\ {\color{Blue} A(4)=0.7756}$

c) When half the dose remains,

$A(t_{1/2})=7e^{-0.55(t_{1/2})}\\ 7/2=7e^{-0.55(t_{1/2})}\\ t_{1/2}=\frac{\ln 2}{0.55}\\ {\color{Blue} t_{1/2}=1.2603 \textup{ hours}}$