Expert: Ahmed Salami - 12/6/2009

Question
The concentration of a drug in the blood stream "t" hour after being administered is modeled by C(t)=120te^-0.4t, where C is in milligrams per cubic centimeter and 0≤t≤24. When is the concentration at its maximum level?  - Can you show the work so I know how to do it? Thanks!-

Answer
Hi Melissa,
I hope you're already familiar with elementary differentiation and the product rule.
C(t)=120t.e^(-0.4t)
The optimum concentration level occurs when dC/dt = 0
dC/dt = 120[t.(-0.4)e^(-0.4t) + 1.e^(-0.4t)]
     = 120[(-0.4t)e^(-0.4t) + e^(-0.4t)]
     = 120[e^(-0.4t)](1 - 0.4t)
Equating to zero,
120[e^(-0.4t)](1 - 0.4t) = 0
e^(-0.4t) is never equal to zero and so we're left with the only option that
1 - 0.4t = 0
1 = 0.4t
t = 1/0.4
 = 2.5 hrs

You can always get back to me.

Regards