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

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

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

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.

Problem

Ornithologists have determined that some species …

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

Heather Zimmers

Caleb Elmore

Samuel Hannah

Calculus 2 / BC Courses

Recommended Videos

Watch More Solved Questions in Chapter 4

Video Transcript

James Stewart

Calculus: Early Transcendentals

Catherine Ross

Heather Zimmers

Caleb Elmore

Samuel Hannah

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

Calculus: Early Transcendentals

Catherine Ross

Heather Zimmers

Caleb Elmore

Samuel Hannah

James StewartCalculus: Early Transcendentals

Catherine Ross

Heather Zimmers

Caleb Elmore

Samuel Hannah

James Stewart

Calculus: Early Transcendentals