Like
Report
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.
Answer
a) $C\left(\frac{a-b \cot \theta}{r_{1}^{4}}+\frac{b \csc \theta}{r_{2}^{4}}\right)$
b) So $\cos \theta=\frac{r_{2}^{4}}{r_{1}^{4}}$ occurs at the minimum.
c) $\theta \approx 79^{\circ}$
Discussion
You must be signed in to discuss.
Top Calculus 2 / BC Educators
Catherine R.
Missouri State University
Anna Marie V.
Campbell University
Heather Z.
Oregon State University
Watch More Solved Questions in Chapter 4
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
Top Calculus 2 / BC Educators
Grace H.
Numerade Educator
Catherine R.
Missouri State University
Anna Marie V.
Campbell University
Heather Z.
Oregon State University
