   Chapter 3.10, Problem 40E

Chapter
Section
Textbook Problem

When blood flows along a blood vessel, the flux F (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel:F = kR4(This is known as Poiseuille’s Law; we will show why it is true in Section 8.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow.Show that the relative change in F is about four times the relative change in R. How will a 5% increase in the radius affect the flow of blood?

To determine

To show: The relative change in F is the four times the relative change in R and to find how a 5% increase in the radius affect the flow of blood.

Explanation

Let F denote the flux and R denote the radius.

It is given that the flux F is proportional to the fourth power of the radius R of the blood vessel that is F=kR4.

The differential is defined by the equation, dF=f(R)dR.

Obtain the derivative of the function f(R)=kR4 as follows,

f(R)=ddR(kR4)=kddR(R4)=k(4R3)=4kR3

Substitute f(R)=4kR3 in the equation for the differential dF=f(R)dR as,

dF=4kR3dRdFdR=4kR3

When ΔR is small the above equation becomes,

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