Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

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

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

JH
J Hardin
Numerade Educator

Like

Report

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

Related Topics

Sequences

Series

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:59

Series - Intro

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

Video Thumbnail

02:28

Sequences - Intro

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

Join Course
Recommended Videos

09:46

Antibiotic pharmacokinetic…

0:00

Antibiotic pharmacokinetic…

05:01

A doctor prescribes a 100 …

04:59

doctor prescribes 100-mg a…

04:32

12. A doctor prescribes 10…

06:09

doctor prescribes 200 mg a…

05:21

A patient takes 150 mg of …

01:02

Use this scenario: A docto…

03:49

The rate at which a drug l…

04:07

Drug pharmacokinetics A pa…

01:55

Solve each problem. See Ex…

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.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
85
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
53
Hosted by: Alonso M
See More

Related Topics

Sequences

Series

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Michael Jacobsen

Idaho State University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:59

Series - Intro

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

Video Thumbnail

02:28

Sequences - Intro

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

Join Course
Recommended Videos

09:46

Antibiotic pharmacokinetics A doctor prescribes a 100-mg antibiotic tablet to b…

0:00

Antibiotic pharmacokinetics A doctor prescribes a 100-mg antibiotic tablet to b…

05:01

A doctor prescribes a 100 -mg antibiotic tablet to be taken every eight hours.…

04:59

doctor prescribes 100-mg antibiotic tablet to be taken every eight hours Just b…

04:32

12. A doctor prescribes 100-mg antibiotic tablet to be taken every eight hours.…

06:09

doctor prescribes 200 mg antibiotic tablet to be taken every hours. Just before…

05:21

A patient takes 150 mg of a drug at the same time every day. Just before each t…

01:02

Use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug tha…

03:49

The rate at which a drug leaves the bloodstream and passes into the urine is pr…

04:07

Drug pharmacokinetics A patient takes 200 $\mathrm{mg}$ of a drug at the same t…

01:55

Solve each problem. See Examples 1–8. Drug Dosage Certain medical conditions ar…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started