Question

The concentration C (in mg/dl), of an antibiotic in a patient’s bloodstream per hour, t, is given by:

Answer 1

We want the concentration of the antibiotic to be at a minimum of 5 mg/dL, so we want to find the time interval, where $t>0$, for which $C(t)\ge5$.

We have

$\dfrac{55t}{t^2+25}\ge5\iff55t\ge5t^2+625\iff t^2-11t+25\le0$

Completing the square, we get

$t^2-11t+\dfrac{121}4-\dfrac{121}4+25\le0\iff\left(t-\dfrac{11}2\right)^2\le\dfrac{21}4$

$\implies\left|t-\dfrac{11}2\right|\le\dfrac{\sqrt{21}}2$

$\implies-\dfrac{\sqrt{21}}2\le t-\dfrac{11}2\le\dfrac{\sqrt{21}}2$

$\implies\dfrac{11-\sqrt{21}}2\le t\le\dfrac{11+\sqrt{21}}2$