Choose the one alternative that best completes the statement or answers the question.
-The concentration $\mathrm { C }$ (arbitrary units) of a certain drug in a patient's bloodstream can be modeled using $C ( t ) = \frac { t } { ( 0.423 t + 2.366 ) ^ { 2 } }$, where $t$ is the number of hours since a 500 milligram oral dose was administered. Using the TABLE feature of a graphing utility, find the time at which the concentration of the drug is greatest. Round to the nearest tenth of an hour.
A) $5.6$ hours
B) $6.4$ hours
C) $7.1$ hours
D) $7.9$ hours

Question

Quizplus · Accepted Answer

The concentration of the drug is greatest when the derivative of $C(t)$ is equal to 0. Using the quotient rule, we have:$$C'(t) = \frac{(0.423t+2.366)^2 - t(2)(0.423)(0.423t+2.366)}{(0.423t+2.366)^4}$$Setting $C'(t) = 0$ and solving for $t$, we get:$$(0.423t+2.366)^2 - t(2)(0.423)(0.423t+2.366) = 0$$$$(0.423t+2.366)^2 - 2(0.423)t(0.423t+2.366) = 0$$$$(0.423t+2.366)^2 - 0.846t^2 - 1.966t = 0$$$$0.846t^2 + 1.966t + 2.366^2 = 0$$$$t = \frac{-1.966 \pm \sqrt{1.966^2 - 4(0.846)(2.366^2)}}{2(0.846)} = -5.6, 6.4$$Since $t$ must be positive, the time at which the concentration of the drug is greatest is $6.4$ hours.

Choose the One Alternative That Best Completes the Statement or Answers