Problem 34 Hard Difficulty

After alcohol is fully absorbed into the body, it is metabolized with a half-life of about 1.5 hours. Suppose you have had three alcoholic drinks and an hour later, at midnight, your blood alcohol concentration (BAC) is 0.6 mg/mL.

(a) Find an exponential decay model for your BAC $ t $ hours after midnight.
(b) Graph your BAC and use the graph to determine when you can drive home if the legal limit is 0.08 mg/mL.

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03:46

Clarissa N.

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Video Transcript

all right, We're doing this problem about blood alcohol concentration, and the first thing we want to do is use the given information to find the exponential decay model. So the given information is that the person starts with a 0.6 milligrams per milliliters, blood alcohol concentration. And we know the half life is 1.5 years now with all the other half life problems we've done so far, they've all fit into this kind of bold where we have a starting amount, why not an ending amount? Why? And we have the half that stands for cutting the substance in half to the end power where n is the number of times it's been cut in half the number of have ings or have ings. However, you'd like to say that and the way we find the number of have ings is we take the amount of time that has elapsed and divided by the half life. For example, suppose that three hours had elapsed three divided by 1.5 would be too. So that tells us it would cut in half two times. So if we take this general equation and we substitute our specific starting amount into it. We get our model that we're going to use for this problem. Okay, now that we have that model weaken, graph it and we can figure out when you can drive home if the legal limit is 0.8 milligrams per millimeter. So we grabbed the calculator and we type in the function 0.6 times 0.5 to the X over 1.5 power, and we also type in the function y equals point await that will give us a horizontal line at a height of point of weight, and we'll let that represent that legal limit. So for a window I have chosen to go from X equals negative 1 to 10 and from y equals negative 0.12 point five and those numbers can be different. You just fiddle around until you find something that shows you a good view of your graph. So the exponential decay equation shows us the blood alcohol concentration level in blue and in red. We see the 0.0, a level and we want to find the point of intersection of those two so we can go into the calculate menu. Choose number five Intersect and then put the cursor on the first curve. Press enter. Put the cursor on the second curve, press enter and move over toward the intersection. Point press enter. And here we see at the bottom of the screen that is telling us. 4.36 Comma 0.8 is the intersection point. So what that means is 4.36 hours will elapse until the blood alcohol concentration is 0.8 4.36 hours since midnight. That's about for something in the morning.