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JH
Numerade Educator

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Problem 70 Hard Difficulty

A patient is injected with a drug every 12 hours. Immediately before each injection the concentration of the drug has been reduced by $ 90\% $ and the new dose in increase the concentration by 1.5 mg/L.
(a) What is the concentration after three doses?
(b) If $ C_n $ is the concentration after the $ n $th dose, find a formula for $ C_n $ as a function of $ n. $
(c) What is the limiting value of the concentration?

Answer

(A). 1.665 $\mathrm{mg} / \mathrm{L}$
(B). $C_{n}=\frac{5}{3}\left[1-0.1^{n}\right]$
(C). $\frac{5}{3} \mathrm{mg} / \mathrm{L}$

Discussion

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Video Transcript

so patient is injected every twelve hours and then immediately before the injection, The concentration of the drug and the body has been reduced by ninety percent. And then the new does. An increase is about one point five milligrams. So here and if we want a picture for this, it looks like for letting see end equal the concentration. So here we have our units milligrams for leader. This is after n doses. Now let's go ahead and just draw a graph. Here we start off with no doze and then after the first those, we go upto one point five and then after that we decrease by ninety percent over the next twelve hours. But then right at the hour, we have another increase. So C one would be one point five c two and be well right before we get Teo the second twelve hour period the end of the period. Well, we'll have reduced by ninety percent, so that means that we'll have ten percent left a previous those of previous concentration So ten per cent left That's just point one or just one over ten. So see two is just see one overton But then we also have two. Added the increase one point five. So this is just one point five over ten plus one point five So one point five plus point one five and that's one point six five that c too. And then we'LL do this one more time, so I'd see to. If we want to draw that on the graph, you could see that it doesn't jump. And then once again from the twenty four hours to the thirty six hour period. So the third twelve hour andro interval will decrease by ninety percent and then once again, well increased by one point five. So see three is just ten percent of the previous concentration, plus one point five. So by our answer over here, that will just be point one six five plus one point five. So that's one point six six five. That will be the answer for Party A. That's the concentration in the body, right after three doses. So there's there's where it is, graphically c three and disappear, or why values, which is CNN. So now let's go on to party CNN. As we've noted, concentration in the body after that does it's kind of formula for CNN, but this time is just a function of end in part A. We've kind of been using the formula. Let me go to the next page. CNN equals C and minus one over ten plus one point five. This was the formula that I was using and party to find the concentration for the first three values. It's given by the fact that before the next the new period, we'd have on ly ten percent of the previous oceans. But then we also increased by the one point five each time. But now, in part B, they want a formula that one a formula to depend on and on Lee. So not not CNN or C N minus one, etcetera. So what we can do it is just keep following this pattern. So if I use this formula again, I can replace see one, see if you see me see and minus one with this term up here and then I divide that by ten, and I add the one point five. So let me call the recording their here. These are the same and then these are the same, and that could go ahead and re write this out. So perhaps we could just keep following this pattern, then end times. That depends on end. So we should end up with a formula that only depends on end. So let's just king following this pattern if we keep following this pattern, Well, what eventually will get down to see one? But what should the denominator b will notice that the's terms right here add up to end. So if I have a one here that I should have it and won my n minus one down there so that they add up to end and then we can see what looks like a geometric series one point five one point five over ten. That should not have been intense. Where that should have been just a ten. Sorry about that. Let me go ahead and erase that. Oops. So if I keep going in this pattern, how many double the power of Cindy? So we started off with n then down here it goes to let's see here and minus two. So it's always one less than the power of ten over here under the sea n. So if this is an minus one, then This must be one less than that. So and minus two. Now, I can go ahead and use the fact that C one is just one point five after the first dosage so I could go ahead and re write. This is one point five and then I can see that the sum is just one point five one point five over ten, all the way up to one point five over ten to the end, minus one. Yeah, so we have all the powers. So now this is a geometric series. Let me. I'm running out of room here. So let me go in and write this on the next page. So this is a fine a geometric Siri's and it can be simplified now, but let there's no need to it'LL actually be easier. And so this is party. Let's just leave that as it is, it can be simplified. There is a formula for finance geometric series, but there's just no need to use it here. We can see the right hand side only depends on end, and we can actually use this The answer part. See? So for Farsi, they wanted the limit of CNN, so Let's go ahead. And if I write, take the limit of this as N goes to infinity. Well, that's just this infinite geometric some and we have an easier formula for that. This is why I would rather just go to infinity first and then use the formula because now we know the formula is first term over one minus our common ratio. So here, that's our first term. And R R is equal to what are we multiplying by each time? Just won over ten. So we'LL have one point five overnight over ten. So that's fifteen over nine, and we can't even simplify that five over three. So that's our answer for Parsi. That's the limiting value of the concentration after and values, and that's your final answer.