Problem 69 Hard Difficulty

A doctor prescribes a 100-mg antibiotic tablet to be taken every eight hours. Just before each tablet is taken, $ 20% $ of the drug remains in the body.
(a) How much of the drug is in the body just after the second tablet is taken? After the third tablet?
(b) If $ Q_n $ is the quantity of the antibiotic in the body just after the $ n $th tablet is taken, find an equation that expresses $ Q_{n + 1} $ in terms of $ Q_n. $
(c) What quantity of the antibiotic remains in the body in the long run?

Answer

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Video Transcript

doctor prescribes one hundred milligrams of antibiotic to be taken every eight hours. And just before each tablet has taken, they should be twenty percent. That's a twenty percent. They're not twenty milligrams. Twenty percent of the drug remains in the body. So for part A. Well, before we get into that, let's actually just draw a picture here of what's going on. You have your you get your injection. So you have one hundred milligrams. Eight hours later, you're down to twenty percent, which is just twenty. And then you get another dosage at the eight hour point. So you'd shown back up to one hundred. So now you add a hundred. So you're at one. Twenty and then you wait until you get to eight hours later. Twenty percent of that, then you do another injection and so on. So now for part A here. How much of the drug is in the body just after the second tablet is taken? So this is when you take the first time. But you have one hundred and then you take your second tablet and we see that we are at one twenty. So how did we get there? We just did one hundred plus twenty, but the word of the twenty come from. That's when he came from taking point two of the original one hundred. So the answer there will just be one twenty and then after the third tablet. So then again. So here we hit were at one twenty. Then we come down to twenty percent. Then we take our third tablet. So this is first, second tablet third and right after we take that, we shoot back up here by one hundred. So what's the total amount? Well, it's just going to be the one hundred that we adhere, plus the twenty percent of the remaining quantity that we have, which was one twenty from our previous answer. So go ahead and just simplify this and this is just one twenty four. So that's after the third tablet and the ones one he was after the second tower, So one hundred after the first, once one day after the second and then up here at one twenty four after the third. So now for party, they're letting Q and be the amount of in a panic in the body after the inn tablet. So here we can see like this is Kyu won que tu and then the one twenty four from the previous power sq three and so on. Let's find an equation for Q and plus one in terms of Q, and we could kind of see how to do this from part A. So the answer here. So this is after you get the dosage of one hundred. This is where they're won. Hundreds coming from. But what are you adding the hundred to each time you're adding it to the twenty percent from your previous dosage? So this will be point two of your previous those suits Q and A. And this is kind of the formula that we used in directly in part. A. This is where the point two, so this is basically impart a we did Q two equals point two kyu won and Cue three equals point two. Cute, too. And let's see if we could throw that lot that part c on the same page here. So I'm going to squeeze that in what quantity of the antibiotic remains in the body in the long run. So basically here we want to know what's the limit of qn. If you continue to take the's, injection these tablets every eight hours. What's that? Your cue? One going to approach. So here we have one hundred and it jumped all the way to one twenty. But then it doesn't jump quite aside on one twenty four. So maybe this dosage is not going to infinity. It's probably going to just level off somewhere. So he had a big, big jump slow down then. Well, we can see that it's just going to continue to slow and slow is slow down. Let's find the actual limit that it approaches. So the way to find this limit is to use part B and take the limit as and goes to infinity on both sides of this equation. Then, since this end, this is on ly differing by from end by one thes to limits are the same. So if you're saying, for example, if the note this limit by L. Than in Part B, if we plug in the limit on each side, we have l equals one hundred plus point to L. Because again, these two quantities qn in Q n plus one have the same limit. Same limit and which were denoting by L. So plugging in El and part B to both sides, we have this and then just solve this for l. This is just the linear equation. You have a point, a l equals one hundred. And then just divide both sides by pointy and you get l equals one twenty five milligrams. So this is the limiting decision, the long run if we continue to use the formula Party and that's our final answer.