Question

Poiseuille’s law asserts that the speed of blood that is rr

centimeters from the central axis of an artery of radius RR is SS(rr) = cc(RR2 − rr2), where cc is a

positive constant. Where is the speed of the blood greatest?

Answer 1

We have studied the exercise and we can see that the speed of blood in an artery can be calculated by Poiseuille's law which is given by

 $S(r)=c\left(R^{2}-r^{2}\right)$

Here R is the radius of an artery; r is the distance from the central axis of an artery and c is a positive constant.

We have to find the value of r where the speed of the blood is greatest.

The value of r cannot be greater than R or less than zero, so r lies in the interval $0 \leq r \leq R.$

Hence the goal is to find the absolute maximum of S(r) within the interval $0 \leq r \leq R.$

We have to find the derivative of S(r) which is denoted by $S^{\prime}(r)$

 $\begin{aligned} S(r) &=c\left(R^{2}-r^{2}\right) \\ S^{\prime}(r) &=c(-2 r) \\ &=-2 c r \end{aligned}$

The critical numbers from $S^{\prime}(r)$ is

 $\begin{array}{r} S^{\prime}(r)=0 \\ -2 c r=0 \\ r=0 \end{array}$

The critical number r=0 lies in the interval $0 \leq r \leq R$

So, we will compute S(r) at r=0 and at the end points r=0 and r=R

 $\begin{aligned} S(0) &=c\left(R^{2}-(0)^{2}\right) \\ &=c R^{2} \\ S(R) &=c\left(R^{2}-(R)^{2}\right) \\ &=0 \end{aligned}$

Therefore, we can conclude that at r=0, speed of the blood is greatest.