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

Ornithologists have determined that some species …

05:38

Question

Answered step-by-step

Problem 80 Hard Difficulty

The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille's Laws gives the resistance $ R $ of the blood as
$$ R = C\dfrac{L}{r^4} $$
where $ L $ is the length of the blood vessel, $ r $ is the radius, and $ C $ is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) THe figure shows a main blood vessel with radius $ r_1 $ branching at an angle $ \theta $ into a smaller vessel with radius $ r_2 $.
(a) Use Poiseuille's Law to show that the total resistance of the blood along the path $ ABC $ is
$$ R = C\left( \dfrac{a - b \cot \theta}{r{_1}^4} + \dfrac{b \csc \theta}{r{_2}^4} \right) $$
where $ a $ and $ b $ are distances shown in the figure.
(b) Prove that this resistance is minimized when
$$ \cos \theta = \dfrac{r{_2}^4}{r{_1}^4} $$
(c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is two-thirds the radius of the larger vessel.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

WZ
Wen Zheng
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Wen Zheng

Numerade Educator

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

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 4

Applications of Differentiation

Section 7

Optimization Problems

Related Topics

Derivatives

Differentiation

Volume

Discussion

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

Missouri State University

Heather Zimmers

Oregon State University

Caleb Elmore

Baylor University

Samuel Hannah

University of Nottingham

Calculus 2 / BC Courses

Lectures

Video Thumbnail

04:35

Volume - Intro

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

Video Thumbnail

06:14

Review

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

Join Course
Recommended Videos

05:28

The blood vascular system …

06:19

The blood vascular system …

09:06

The blood vascular system …

06:54

Blood flow The resistance …

01:03

Blood is carried throughou…

07:00

Blood Flow The figure show…

03:37

Vascular Branching A small…

02:30

The flow of blood in a blo…

03:27

Blood flow The shape of a …

06:47

Vascular Branching A small…

18:29

Blood contains positive an…

04:18

Blood Flow As blood moves …

03:44

Blood Flow As blood moves …

01:29

The flow of blood in a blo…

03:49

The flow of blood in a blo…

Watch More Solved Questions in Chapter 4

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

Video Transcript

The problem is the blood was muscular. System consists off blood y sauce that come way Lord from the heart into the organs on the back to the heart. This this time should work so as to minimise energy. Undated By the heart pumping the blood in particular, this energy is reduced when the resistance of the broads is lower. One ofthe possibly snores gives the resistance are off the blood as art is equal to C have our overarching power off for while I r i all his land off the blood of my soul Art is the gliders and see it's a positive cast him to determine the puzzle will last Watson closet e off Brandt. Now let's look at it. It's a figure. It's a figure shows my implied Why so always writers are one brunching at an angle greater into a smaller wife. So wizard like there's hot too. The problem part a use parcel is law to showed that the total resistance out the blood long past ABC It is our secret state hams implant. Speak attendant data over our ones to the party for plus being have seen Costigan over are too is a power off for? For part A. The puzzle is law is ill. Our physical to sleep. L over onto the power ofthe war. Now let's look at this graph we can see. Maybe is the control eh? Minus it. Hams contend data we have to be. If they could train one speed, have school attendant on the PC is there too. He hams, Costigan data. Then we use the puzzle is small we have Are we going to see tams aim, honesty, contentment or are one to the public for us? Three times cause Sigmund over to the pub ofthe war. But be proof that this resistance is minimized One consigned, paid, Iet's go to I'm chiu to the parliament for over one to the power ofthe war Take curative yard. This is Echo Two faint house B R O r. One to the power wealth or tam's coincident. They had a square minus. You know where on too? To the party of four Cho Seon Bitter times Could attendant Yeah, this's equal to save me because Second there, squire, Over one two, the part ofthe war House one minus. I want to the power off or oh, are two to the power off or could tangent data or who's second? No. But here Crittendon Data or Costigan bidder is equal to who signed data. If we like we are. If there is a country zero, we have who signed, It's the truth are one to two to the power off or or want to the power ofthe war. So when theta is greater than cosigned, a worse to two parts off or well, I want the power off or we have already begun is, which is on cereal and thinner is more of a this number. And he already did. Eyes smaller than cereal on behalf. Why is the country cause I'LL co sign out? Sure to the power for over outline to the parole for or Whose scientific Isaac go to this number? Our has some minimal Why Do party find is up to no branching angle When the riders off the smaller blood, Why so is too thers? The writer's outside larger Wiesel. So we have two over one. If they wanted to third so call sign, they got to third the power of four. It is a part seventy nine degrees

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
142
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Derivatives

Differentiation

Volume

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Heather Zimmers

Oregon State University

Caleb Elmore

Baylor University

Samuel Hannah

University of Nottingham

Calculus 2 / BC Courses

Lectures

Video Thumbnail

04:35

Volume - Intro

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

Video Thumbnail

06:14

Review

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

Join Course
Recommended Videos

05:28

The blood vascular system consists of blood vessels (arteries, arterioles, capi…

06:19

The blood vascular system consists of blood vessels (arteries. arterioles, capi…

09:06

The blood vascular system consists of blood vessels (arteries, arterioles, cap…

06:54

Blood flow The resistance to blood flow in the circulatory system is one measur…

01:03

Blood is carried throughout the body by the vascular system, which consists of …

07:00

Blood Flow The figure shows the blood flow in a small artery of the body. The …

03:37

Vascular Branching A small blood vessel of radius $r$ branches off at an angle …

02:30

The flow of blood in a blood vessel is \intaster toward the center of the vesse…

03:27

Blood flow The shape of a blood vessel (a vein or artery) can be modeled by a c…

06:47

Vascular Branching A small blood vessel of radius $r$ branches off at an angle …

18:29

Blood contains positive and negative ions and thus is a conductor. A blood vess…

04:18

Blood Flow As blood moves through a vein or an artery, its velocity $v$ is grea…

03:44

Blood Flow As blood moves through a vein or an artery, its velocity $v$ is grea…

01:29

The flow of blood in a blood vessel is \intaster toward the center of the vesse…

03:49

The flow of blood in a blood vessel is faster toward the center of the vessel a…
Additional Mathematics Questions

02:07

One of the base angles of an isosceles trapezium is 80 degree. How will we t…

02:27

Purchased machinery rs 95000 and paid rs 5000 for freight jonural entry

04:08

Rohan has written some kpis to help his fitness centres achieve the business…

03:40

Rahul has 260 coins of rs 1,rs 2 ,rs 5 altogether. The total value of the mo…

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