Answer to Question #272383 in Calculus for fatima mazhar

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.