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Problem 69 Hard Difficulty

After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream (blood alcohol concentration, or BAC) surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The function

$$ C(t) = 1.35te^{-2.802t} $$

models the average BAC, measured in mg/mL, in a group of eight male subjects $t$ hours after rapid consumption of $15$ ml of ethanol (corresponding to one alcoholic drink). What is the maximum average BAC during the first $3$ hours? When does it occur?

Source: Adapted from P. Wilkinson et. al., "Pharmacokinetics of Ethanol after Oral Administration in the Fasting State," Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207-24.

Answer

$B A C_{\max }=0.177244 \frac{ \mathrm{mg}}{\operatorname{mL}}$ ; $t\operatorname{_{max}}=\frac{1}{2.802}$

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Video Transcript

here we have a function C. F. T equals 1.35 times T times E. To the negative 2.8 oh two times T. Power. Uh C. Represents the blood alcohol concentration. Uh And it is a function of t uh The number of hours. And we want to find uh what is the maximum blood alcohol concentration during the first three hours? And exactly at what value of T. Uh You know, what exact number of hours does it occur? So here is C. As a function of T. And uh for uh the interval between zero and three the first three hours, we want to find out when this function, this blood alcohol concentration function has a maximum. Uh So we could do it two ways. You could find the derivative of this function with respect to T. Set that derivative equal to zero and solve that equation. 40. And the value of T. Uh That makes us the first derivative equal to zero. Um At that value of T. Where the first derivative become zero. You have either a local maximum or local minimum. So this is one way where you can find uh the maximum value of the uh c. Of T function by setting the first derivative equal to zero. Um We are going to take a graphing approach. So we're going to use dez most graphing calculator input. This function and look for the maximum on the graph. So here is the blood alcohol concentration function uh entered. And this blue curve is the graph of the blood alcohol uh concentration to cfx function. You can see in the end of role. Um Now the only difference was I had to use using Dismas instead of using T. For hours. I'm using the variable X. For hours. But you can see on the end of all, between zero hours and three hours, our sea of X function does reach a maximum point right here. Um So during the first three hours the blood alcohol concentration does reach a maximum the time. Okay? The X coordinate the T coordinate at T equals 0.357 hours. Uh Our function reaches a maximum at T equals 0.357 hours. Uh We have the maximum blood alcohol concentration and that maximum Blood alcohol concentration is .177. And that is measured in milligrams per mm. So at .357 hours the blood alcohol concentration reaches a maximum a .177 milligrams per milliliters.

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