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

One side of a right triangle is known to be $ 20 …

01:13

Question

Answered step-by-step

Problem 37 Hard Difficulty

(a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height $ h, $ inner radius $ r, $ and thickness $ \Delta r. $
(b) What is the error involved in using the formula from part (a)?


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Linda Hand
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Linda Hand

Numerade Educator

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

More Answers

01:32

Amrita Bhasin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 10

Linear Approximation and Differentials

Related Topics

Derivatives

Differentiation

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
Recommended Videos

03:38

$$ \begin{array}{l}{\text …

04:07

(a) Use differentials to f…

02:40

(a) Use differentials to f…

03:45

The radius $r$ of a sphere…

03:15

The volume of a right circ…

09:07

Volume and Surface Area Th…

10:50

The circumference of a sph…

08:08

$$ \begin{array}{l}{\text …

04:53

The circumference of a sph…

Watch More Solved Questions in Chapter 3

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

Video Transcript

here I was, Cylinder. And if I want to find its volume, all I have to do is find the area of the base and multiply it by the heights. So that would be pi r square area of the base times the height. Okay, so now what I'm gonna do is I'm gonna just poor Cem gold or whatever around the outside and make a shell around it in the shell still has the same height age. But now it's radius is different. So let's say the thickness of the shell is delta are Okay, Well, all we have to do to find the volume of the shell iss using differentials take the differential both sides. So volume of the shell is approximately d v. They change in volume, which would be pi times the derivative of R squared, which would be to r D. R or Delta are time to age because H isn't changing. So two pi r h Delta are Okay, So that's a formula we could use to approximate the volume of a shell if we knew the inner radius and then the thickness of the shell. So how much error are we making? All right. Let me draw you a better one was a little bit more detailed. All right, so this is still a judge, and then ours from the center to the inside circle. And then the thickness here is Delta are Okay, so let's find the actual volume. So what I'm gonna do is I'm gonna take my magic scissors here, and I'm going to cut it open. Uh huh. Okay, because it has this in there. I'm going to cut it open, and I'm gonna unfold it. Let's do some color coding. Okay, here's the inside circles. So that's gonna be this? Yeah. And then, Oops. Let's make the outside circle blue. And it's bigger than the inside circle by Delta are. And then the thickness right here. That would be this. And this and then the height. Okay, So what I have here is not exactly a box because, well, it's a box, but it's not rectangular box because the side here is a trapezoid. So to find the volume of this, which is the volume of the shell, I have to do area of the base times the height. Okay, so here's the base I'm talking about right here. So it's a trapezoid. Let's put some stuff on you. Well, if the radius of the inner circle is our than the length of the inner circle is two pi r Can The radius of the outer circle is our plus Delta are so its length to pie R plus Delta are And then this thickness red here, remember, is delta are Okay, So the area of a trapezoid half big base to buy our plus delta are a plus. Small base times the height, which is delta are height of the trapezoid approximately. Yeah. Sorry, I Drew. This is Delta are right here all of these times the height of the, um, box here. So H so that simplifies into both These have a two pi. So let's pull that out. R plus Delta R plus R Delta R h so pi to R plus Delta are Delta R h. So that's the actual volume of the shell. So then the error we're making is our approximate volume minus the actual volume. Okay, The approximate volume two pi r h delta are hi r h don't are minus the actual volume pie to our plus delta are Delta R h. Okay. Okay. They both have an H. Aiken. Factor out. They both have a pie. I can factor out. And they both having no, I doubt a are that I convict her up. So pi age Delta R And then what's left to our minus parentheses e to r plus Delta are so pi Age Delta are Maya's delta are times minus delta are or pi h delta R squared. I got minus for the answer because maybe I was supposed to subtract him the other way, and then that'll make it plus Okay, so that's the error that we're making, which, if delta R is small, this will be a very small error. Okay, there you go.

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
192
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
82
Hosted by: Alonso M
See More

Related Topics

Derivatives

Differentiation

Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Derivatives - Intro

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

Video Thumbnail

44:57

Differentiation Rules - Overview

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

Join Course
Recommended Videos

03:38

$$ \begin{array}{l}{\text { (a) Use differentials to find a formula for the app…

04:07

(a) Use differentials to find a formula for the approximate volume of a thin cy…

02:40

(a) Use differentials to find formula for the approximate volume of tnin cylind…

03:45

The radius $r$ of a sphere is measured to be $22.7 \mathrm{cm} \pm 0.2 \mathrm{…

03:15

The volume of a right circular cylinder is given by $V(r, h)=\pi r^{2} h .$ Fin…

09:07

Volume and Surface Area The radius of a spherical balloon is measured as 8 inch…

10:50

The circumference of a sphere was measured to be $ 84 cm $ with a possible erro…

08:08

$$ \begin{array}{l}{\text { The circumference of a sphere was measured to be } …

04:53

The circumference of a sphere was measured to be 84 $\mathrm{cm}$ with a possib…

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