# In the equation  for concentration of medicine in bloodstream    C(t)=3t//05t^2+2.     After how many minutes will the maximum amount of medication be in the bloodstream? What is the max of the medicine that will be in body before it decreases?  would this max rep a hor asymptote?

You need to solve an optimization problem, hence, you need to use derivatives to evaluate after how many minutes the maximum amount of medication will be in the bloodstream.

Differentiating the given concentration function with respect to t, yields:

`C'(t) = ((3t)//(0.5t^2+2))'`

`C'(t) = ((3t)'(0.5t^2+2) - (3t)(0.5t^2+2)')/((0.5t^2+2)^2)`

`C'(t) = (3(0.5t^2+2) - (3t)(t))/((0.5t^2+2)^2)`

`C'(t) = (1.5t^2+2 - 3t^2)/((0.5t^2+2)^2)`

`C'(t) = (-1.5t^2+2)/((0.5t^2+2)^2)`

You need to solve the equation `C'(t) = 0` , such that:

`(-1.5t^2+2)/((0.5t^2+2)^2) = 0 => -1.5t^2+2 = 0 => t^2 = 2/1.5 => t ~~ 1.154.`

Hence, evaluating the time t for the maximum amount of medication will be in the bloodstream yields ` t ~~ 1.154` units of time (hours).

Approved by eNotes Editorial Team