Question

the concentration of a drug (in parts per million) in a patients bloodstream t hours after administration of the drug is given by the function p(t)=-t^{4}+12t^{3}-58t^{2}+132t. the corresponding graph is shown to the right. complete parts a through d.
a. use the formula to determine when the drug will be totally eliminated from the bloodstream.
the drug will be totally eliminated from the bloodstream in hours.
(simplify your answer.)

Explanation:

Step1: Set the function equal to 0

We want to find when $P(t)=0$, so we set $-t^{4}+12t^{3}-58t^{2}+132t = 0$. Factor out $-t$: $-t(t^{3}-12t^{2}+58t - 132)=0$.

Step2: Try to find roots of the cubic

We can test some simple values of $t$. By the rational - root theorem, possible rational roots are factors of 132. Testing $t = 2$: $2^{3}-12\times2^{2}+58\times2 - 132=8-48 + 116-132=-56
eq0$. Testing $t = 6$: $6^{3}-12\times6^{2}+58\times6 - 132=216-432+348 - 132=0$.

Step3: Factor the cubic

Since $t = 6$ is a root, we can factor $t^{3}-12t^{2}+58t - 132$ as $(t - 6)(t^{2}-6t + 22)$.

Step4: Solve the quadratic

The quadratic formula for $ax^{2}+bx + c=0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For $t^{2}-6t + 22$, $a = 1$, $b=-6$, $c = 22$. Then $\Delta=b^{2}-4ac=(-6)^{2}-4\times1\times22=36 - 88=-52<0$, so it has no real roots.

Step5: Find all real roots

We had $-t(t - 6)(t^{2}-6t + 22)=0$. The real roots are $t = 0$ (corresponds to the time of administration) and $t = 6$.

Answer:

6