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).**

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