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JH
Numerade Educator

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Problem 71 Hard Difficulty

A patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, $ 5% $ of the drug remains in the body.
(a) What quantity of the drug is in the body is in the body after the third tablet? After the $ n $th tablet?
(b) What quantity of the drug remains in the body in the long run?

Answer

a. 157.5
b. $\approx 157.895$

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Video Transcript

the patient takes one hundred fifty milligrams of a drug at the same time every day. And just before the tablets taken, only five percent of the drug remains in the body. And then we'd like to know what quantity of the drugs in the body after the third tablet or after the end tablets. So for part A, we could even draw picture for this. So what? Q N Just be the quantity in body after and tablets or, in other words, after ten days. So the quantity and of course, this is in milligrams. That's the units. I'm a case, those air being stressed. You can always point those out. So here, after the first day, we'LL get one hundred fifty, and then after that we would have decreased all the way down to just only five percent of that. So five percent of one fifty that's just one fifty over twenty fifteen over, too. And then after that, right before we take the second tablet, that's where we're at at fifteen over, too. But then we take another tablet. That's one hundred fifty. So then we add one fifty to this, so that will push us up too. So we have this right here seven and a half, So then we would have jumped up to one fifty seven point five, and then we could do this one more time. Then we would decrease the only five percent of that, and then we would add one fifty to this. So the value so Kyu won. When we take our first tablet, we have one fifty. This value that we just found over here was cute, too, and then Q three will be five percent of this, and then we just add by one hundred fifty. So you have in this case, you can go ahead and simplify this frank and the fractions together and simplified to get your final answer. But that's Q three so otherwise, if a calculator is not needed, just use the fractions. If you have to get the decimal, go ahead, use the calculator. That's just Q three. And then let's go and write on the formula for Q N. That's just the previous concentration in your body, but then you want five percent of that. So you divide by twenty something right in this way. That's five percent of the previous, but then immediately after that, you add by one fifty. So this is the formula for box for second part of party. And then let's go ahead and use this formula to answer Part B. So for party is we'd like to know if we keep doing this day after day in the limit in the long run, with the amount of concentration and the body. So first notice that let's denote this Lim Kyu end by l and notice that we can go ahead and replace the Q end with Q and minus one because his end goes to infinity and minus one also goes to infinity. So it's the same limit. So if we go ahead and take this expression over here, I was going and take the limit of both sides. Then we get L. L. Over twenty plus one hundred fifty saying it. Nineteen over twenty l equals one fifty and then just solve that for L and nineteen is prime, so there should be no cancellation here. This can be approximated just using a calculator. But if we keep it in this fraction form, that's the exact answer. So this will be the amount of concentration that that's in the body in the long run, and that's your final answer