Problem

A patient is injected with a drug every 12 hours.…

09:09

Question

Answered step-by-step

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?

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 2

Series

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

Watch More Solved Questions in Chapter 11

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Problem 75

Problem 76

Problem 77

Problem 78

Problem 79

Problem 80

Problem 81

Problem 82

Problem 83

Problem 84

Problem 85

Problem 86

Problem 87

Problem 88

Problem 89

Problem 90

Problem 91

Problem 92

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.

Get More Help with this Textbook

James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups

Study with other students and unlock Numerade solutions for free.

Problem

A patient is injected with a drug every 12 hours.…

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Catherine Ross

Anna Marie Vagnozzi

Kayleah Tsai

Michael Jacobsen

Calculus 2 / BC Courses

Recommended Videos

Watch More Solved Questions in Chapter 11

Video Transcript

James Stewart

Calculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Kayleah Tsai

Michael Jacobsen

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Kayleah Tsai

Michael Jacobsen

James StewartCalculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Kayleah Tsai

Michael Jacobsen

James Stewart

Calculus: Early Transcendentals