Use the Poiseuille's Law to show that the total resistance of the blood along the path АВС is а. - b cot 0 b csc 0 а — R = C b. Prove that the resistance when dR/d0 = 0 is Cos O %3D .4 Find the optimal branching angle (correct to the nearest degree) when the radius of the maller blood vessel is two thirds of the larger vessel.

Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The blood vascular system consists of blood vessels that convey blood from the heart to the
organs and back to the heart. This system should work so as to minimize the energy
expanded 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
r4
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. The figure below shows a main blood vessel with
radius r, branching at an angle 0 into a smaller vessel with radius r2:
vascular
branching
Use the Poiseuille's Law to show that the total resistance of the blood along the path
АВC is
а.
a – b cot 0
b csc 0
+
R = C
4
b. Prove that the resistance when dR/d0 = 0 is
%3D
cos 0
%3D
.4
Find the optimal branching angle (correct to the nearest degree) when the radius of the
smaller blood vessel is two thirds of the larger vessel.
Transcribed Image Text:The blood vascular system consists of blood vessels that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expanded 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 r4 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. The figure below shows a main blood vessel with radius r, branching at an angle 0 into a smaller vessel with radius r2: vascular branching Use the Poiseuille's Law to show that the total resistance of the blood along the path АВC is а. a – b cot 0 b csc 0 + R = C 4 b. Prove that the resistance when dR/d0 = 0 is %3D cos 0 %3D .4 Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is two thirds of the larger vessel.
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