Blood Flow As blood moves through a vein or an artery, its...

Blood Flow As blood moves through a vein or an artery, its velocity $v$ is greatest along the central axis and decreases as the distance $r$ from the central axis increases (see the figure). The formula that gives $v$ as a function of $r$ is called the law of laminar flow. For an artery with radius $0.5 \mathrm{cm},$ the relationship between $v(\mathrm{in} \mathrm{cm} / \mathrm{s})$ and $r(\mathrm{in} \mathrm{cm})$ is given by the function $$ v(r)=18,500\left(0.25-r^{2}\right) \quad 0 \leq r \leq 0.5 $$ a. Find $v(0.1)$ and $v(0.4)$ b. What do your answers to part.(a) tell you about the flow of blood in this artery? c. Make a table of values of $v(r)$ for $r=0,0.1,0.2,0.3,0.4,0.5$ d. Find the net change in the velocity $v$ as $r$ changes from $0.1 \mathrm{cm}$ to $0.5 \mathrm{cm}.$

Blood Flow As blood moves through a vein or an artery, its velocity $v$ is greatest along the central axis and decreases as the distance $r$ from the central axis increases (see the figure). The formula that gives $v$ as a function of $r$ is called the law of laminar flow. For an artery with radius $0.5 \mathrm{cm},$ the relationship between $v(\mathrm{in} \mathrm{cm} / \mathrm{s})$ and $r(\mathrm{in} \mathrm{cm})$ is given by the function
$$
v(r)=18,500\left(0.25-r^{2}\right) \quad 0 \leq r \leq 0.5
$$
a. Find $v(0.1)$ and $v(0.4)$
b. What do your answers to part.(a) tell you about the flow of blood in this artery?
c. Make a table of values of $v(r)$ for $r=0,0.1,0.2,0.3,0.4,0.5$
d. Find the net change in the velocity $v$ as $r$ changes from $0.1 \mathrm{cm}$ to $0.5 \mathrm{cm}.$

Key Concepts

Table of Values

Creating a table of values is a method of organizing input-output pairs for a function. This aids in visualizing and analyzing the behavior of the function across a range of values, making it easier to identify trends and patterns in the data.

Net Change

The concept of net change refers to the difference in the output of a function over an interval of its input. It is used to quantify the overall increase or decrease in the variable of interest, such as how the velocity of a fluid changes between two specific points.

Quadratic Functions

Quadratic functions, which include a squared term, exhibit a parabolic shape. In the context of flow dynamics, a quadratic relationship can describe how a quantity like velocity varies with distance from a central axis, showing a maximum at one point and decreasing symmetrically.

Function Evaluation

Function evaluation involves substituting specific input values into a function to obtain corresponding output values. This process is essential for determining characteristics of a model, such as calculating the velocity at particular points along a pipe or artery.

Laminar Flow

Laminar flow describes the smooth, orderly movement of fluid in layers, with the highest velocity typically found at the center of the flow and lower velocities near the boundaries. This concept is fundamental in fluid dynamics for understanding how fluids such as blood flow through vessels.

Transcript

00:01 We have been given the relation, the velocity of the blood and the distance it travels.

00:06 So what we get as v is a function of r and it is the law of laminar flow.

00:12 Law of laminar flow.

00:16 So in the question we have been given that v are where v is the velocity and r is the distance.

00:24 So, we get 185 double 0 multiplied by 0 .25.

00:32 2 .5 minus r square and there is a condition that r is varying from 0 to 0 .5.

00:43 And it's basically this basically comes out to be this r is in centimeters.

00:48 It's given in the problem itself.

00:50 Now in the part a we need to find in part a we need to find.

00:54 What is v of 0 .1.

00:56 So we just need to put 0 .1 in the place of r so we get 0 .25 minus 0 .1 square square.

01:06 And no this is not the whole square.

01:09 This square was there only.

01:11 So this becomes now the value of this comes out to be 4 440 and the second thing we have been asked to find the value of v 0 .4 .4.

01:22 We will get to find the value of v 0 .4.

01:23 We will get to find.

01:24 185 -00 0 .25 minus 0 .4 square and this gives the value as 1665.

01:36 Hence these are the two required values.

01:39 Now in the next question it says what what it represents in the part a about the flow.

01:47 So we see that when r is increasing when r is increasing the velocity is decreasing...

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