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Question (8 points)Ethanol decomposesbody by zero-order process[Aol / 2k ; [A] = [Ao] . ktKombucha centuries old beverage that is made when sweetened tea allowed fe...

Question

Question (8 points)Ethanol decomposesbody by zero-order process[Aol / 2k ; [A] = [Ao] . ktKombucha centuries old beverage that is made when sweetened tea allowed ferment in culture of bacteria and veast and can contain up to 1% ethanol by volume Lindsey regularlv drank kombucha for its perceived health benefits but was concerned about her blood alcohol level: When drinking kombucha; Lindsev'$ maximum blood alcohol concentration (BAC) was 0.5 mg/1OOmL of blood. Given that the Initia half-lif

Question (8 points) Ethanol decomposes body by zero-order process [Aol / 2k ; [A] = [Ao] . kt Kombucha centuries old beverage that is made when sweetened tea allowed ferment in culture of bacteria and veast and can contain up to 1% ethanol by volume Lindsey regularlv drank kombucha for its perceived health benefits but was concerned about her blood alcohol level: When drinking kombucha; Lindsev'$ maximum blood alcohol concentration (BAC) was 0.5 mg/1OOmL of blood. Given that the Initia half-life for ethanol In Lindsey'$ body ' 4Sminutes, how long will It take for her BAC to be below detectable levels (<0,01 mg/10dmL)?



Answers

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
$$C(t)=1.35 t e^{-2.802 t}$$
models the average $\mathrm{BAC}$ , measured in mg/mL, of a group
of eight male subjects $t$ hours after rapid consumption of
15 $\mathrm{mL}$ of ethanol (corresponding to one alcoholic drink).
What is the maximum average BAC during the first
3 hours? When does it occur?

In this problem we are given the blood alcohol content and units of milligrams similar leader after two minutes of consumption that E two equals 20.0 to 252.46 17 patients information. We want to answer any which requires us to use differentiation via the chain rule that is a derivative of a function F of X with respect to X. Is F prime G n x n G. Products which is a function inside about seven A. We use the general to find the rate of change of BF 10 minutes first, we find the prime th in the chain rule as 100.2 to 5.46 17 plus. By the product rule 170.467 times 0.0 to +25 to 0.46 17. Thus our solution is for plugging into equals tend to be primed for a. Be prime 10 equals 7.5, 10 to the negative 30 mg per milliliters per minute and b we find the rate of change is 30 minutes. For this we simply forget 30 giving be a third time of 30 equals three times and 93rd milligram milligrams per milliliters per minute.

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.

In this video, we're gonna be covering real numbers and their properties in particular. We're gonna be plugging real numbers into the formula and solving the formula. So we'll take a look at the screen here. I've written down the formula for determining the B. A. C. Which is the alcohol content and the blood. In order to find out the story, you're gonna multiply the total number of ounces that a person has drink times the percentage of alcohol content and each drink Times 0.075, which is going to be a constant, divided by the body weight of the person drinking in pounds minus the total hours that they were drinking times 0.015, which is another constant and formula. So in order to start plugging in numbers, but we're gonna look at the problem and the problem they made for the number of ounces, they may give you the total number of ounces per drink. And to tell you how many drinks someone has. So let's say someone had to Two glasses of wine each top which sits ounces for the wine so much, about two x 6 to get 12. Then they're going to give you the percentage of alcohol content and each drink and you can just plug that percentage straight in. You don't have to change it to a decimal. Just plug it straight into the formula. You're gonna multiply. Bye. The constant. 0.075, divide by the body weight. And they subtract The hours that they were drinking times. 0.015, which is going to be a constant. All right. So now you can just plug this into the calculator, calculate does order of operations for you. But if you do not have a calculator, you have to do a hand by hand. You use the order of operations. Pimp dash, which you see. I have written down here parentheses as opponents modification. Division addition subtraction. So first you do multiplication and division left to right because you have no parentheses or as opponents operations we have parentheses written. But this just represents multiplication. So you would do multiplication and division have to right then we don't have any addition. So then lastly what you would do the subtraction. So if you want to do order of operations, it would look like you. Let's apply all this together, then divide. And this would be 0.084 around it to the thousands place. And then you would do this modification right here. This is going to be 0.030. And this subtraction problem right here is 0.0 5 4 and this is going to be your BSC but now we're going to look at the same problem. And let's say this lady was drinking, it took her four hours to drink two glasses instead of two hours. And how would that how would that affect our B. A C. Level? So we're going to take a look. We're gonna plug in. You still have two glasses. They had six oz of wine on them. The wind still have 14% alcohol concentration. Just the same constant in every formula. Or in every based formula, divided by the same weight minus instead. This time. And this time we're going to have her drinking for four hours and this is going to be the same constant. All right. So it took four hours to drink the same two glasses of wine. How does this affect the B A. C. Level? So I'm gonna break it down just like I did the above problem. Do this order of operations right here. So the same order of operations is going to be 0.084, the same as we have up here because we drink the same amount of alcohol and wear the same amount. But right here we have a change because you drink a little longer. So this is going to be zero zero now And this is going to equal zero 0-4. So as you see, our BAC level is less than our other B A C level because we drank longer, so the longer you drink, you take the drink the same amount of alcohol the last year BSC level would be.

So you've got this equation here. And so first Richard being asked to find out what is the actual percentage after a certain amount of time here. So first in one half an hour or 40 equals one half plugging that in there. So 0.23 times one half you to the negative 0.4 times one half there. That's approximately equal. 2.9 41 percent here. So we have a of eight next. So it's going to be a 0.23 times a day. It's the negative 0.4 times out there. And so that one it's approximately equal to, so it's 0.75 percent there. Okay. And so that would be for that first part then for the second part here. So they're asking us to find the rate of change is. So first to do that, we need to find what a priority is. It's going to be 0.238 the negative 0.4 T here, and then plus 0.23 T. Times negative 0.4 E. To the negative point for tea. And that's by a product rule here. That's how I got that. Okay. And so next we're gonna plug in. See a prime of one half. That's that's equal to point 15 serial six percent here. Oh, that's percent per hour than a prime of eight is equal to approximately negative 0.206 percent power there. All right. Those are your answer, sir?


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