# Thread: A couple more optimization problems

1. ## A couple more optimization problems

Couldn't figure this one out for the life of me... got a few steps down and then it all went downhill from there.

If the concentration of a drug in the bloodstream at time "t" is given by this function,
where a, b(b>a), and k are positive constants that depend on the drug, at what time is the concentration at its highest level?

And the second question:

The motion of a particle is given by s(t) = 5cos(2t + pi/4). What are the maximum values of the displacement, the velocity, and the acceleration?

2. ## I'd help

but I cant read the paper...rewrite it...preferably in LaTeX

3. Originally Posted by NAPA55
Couldn't figure this one out for the life of me... got a few steps down and then it all went downhill from there.

If the concentration of a drug in the bloodstream at time "t" is given by this function,
where a, b(b>a), and k are positive constants that depend on the drug, at what time is the concentration at its highest level?

And the second question:

The motion of a particle is given by s(t) = 5cos(2t + pi/4). What are the maximum values of the displacement, the velocity, and the acceleration?

$\displaystyle c(t)=\frac{k}{b-a}\left[ e^{-at}-e^{-bt}\right]$

Taking the derivative we get...

$\displaystyle \frac{dc}{dt}=\frac{k}{b-a}\left[ -ae^{-at}+be^{-bt}\right]$

setting equal to zero and solving we get

$\displaystyle -ae^{-at}+ be^{-bt}=0 \iff \frac{a}{b}=\frac{e^{-bt}}{e^{-at}}\iff \frac{a}{b}=e^{(a-b)t} \iff ln \left( \frac{a}{b}\right)=(a-b)t$

$\displaystyle t= \frac{ln \left( \frac{a}{b}\right)}{a-b}$