Question

20. Suppose you need to re-administer this injection at the point when the concentration is less than 0.5%. If the first injection was given at 8:00 am, what time should the next injection be given?

Answer 1

Given that the function

$C(h)=\frac{2h^2+5h}{h^3+8}$

models the concentration of a medication in the bloodstream (as a percent), hours (h) after its injection into muscle tissue.

The number of hours it will take for the concentration of the injection to be less than 0.5% is given by:

$0.5\ \textgreater \ \frac{2h^2+5h}{h^3+8} \\ \\ \Rightarrow0.5(h^3+8)\ \textgreater \ 2h^2+5h \\ \\ \Rightarrow h^3+8\ \textgreater \ 4h^2+10h \\ \\ \Rightarrow h^3-4h^2-10h+8\ \textgreater \ 0 \\ \\ \Rightarrow h\ \textgreater \ 5.54$

Given that the first injection was given at 8:00 am, the time the next injection should be given is 2 pm.