College Algebra, Chapter 4, 4.1, Section 4.1, Problem 68

Suppose a a certain drug is taken orally, the concentration of the drug in the patient's blood stream after $t$ minutes is given by $C(t) = 0.06t - 0.0002t^2$, where $0 \leq t \leq 240$ and the concentration is measured in mg/L. When is the maximum serum concentration reached, and what is that maximum concentration?

The function $C$ is a quadratic function with $a = -0.0002$ and $b = 0.06$. Thus, its maximum value occurs when

$\displaystyle t = - \frac{b}{2a} = - \frac{0.06}{2 (-0.0002)} = 150$ min.

The maximum concentration is $C(150) = 0.06(150) - 0.0002 (150)^2 = 4.5$ mg/L.

So the serum will reached maximum concentration is $150$ min. and the maximum concentration of serum is $4.5$ mg/L.