Concentration of drug in blood stream?

Question

Let C(t) be the concentration of drug in bloodstream. As the body eliminates the drug, C(t) decrease at the rate that is proportional to the amount of the drug that is present at the time. Thus dC/dt =-kC(t) where k is positive number. If C(0) is the concentration at time t=0, find the concentration at time t.

help me on this question please.....

Answer 1

dC/dt = -k*C

multiply by dt and divide by C to get

dC/C = -k*dt

Now integrate both sides

∫(1/C)dc = -k∫dt

The left integral is ln(C), and the right is -kt, and add a constant:

ln(C) = -kdt +K

take e to the power of each side:

C = e^(-kt+K)

C = K'e^-kt

Evaluate K' from C(0) at t = 0; set t = 0 and you get

C(0) = K'

So the result is C(t) = C(0)*e^-kt