Problem 73 - Algebra and Trigonometry

Question

As blood moves through a vein or an artery, its velocity ? is greatest along the central axis and decreases as the distance r from the central axis increases (see the figure). The formula that gives ? as a function of r is called the law of laminar flow. For an artery with radius 0.5 cm, the relationship between ? (in cm/s) and r (in 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, \(0.5 .\)

Step-by-Step Solution

Answer

Velocity decreases with distance from the artery's center, consistent with laminar flow.

Step 1: Evaluate v(0.1)

We need to calculate \( v(0.1) \) using the provided function \( v(r) = 18,500(0.25 - r^2) \). Substitute \( r = 0.1 \):\[v(0.1) = 18,500 \times (0.25 - (0.1)^2) = 18,500 \times (0.25 - 0.01)\]Simplify further:\[v(0.1) = 18,500 \times 0.24\]Calculate:\[v(0.1) = 4,440 \, \text{cm/s}\]

Step 2: Evaluate v(0.4)

Now calculate \( v(0.4) \) by substituting \( r = 0.4 \) into the function:\[v(0.4) = 18,500 \times (0.25 - (0.4)^2) = 18,500 \times (0.25 - 0.16)\]Simplify:\[v(0.4) = 18,500 \times 0.09\]Calculate:\[v(0.4) = 1,665 \, \text{cm/s}\]

Step 3: Analyze flow rates

From our calculations, \( v(0.1) = 4,440 \, \text{cm/s} \) and \( v(0.4) = 1,665 \, \text{cm/s} \). This suggests that the blood velocity is higher near the center of the artery (at \( r = 0.1 \)) and decreases as the distance from the center increases (at \( r = 0.4 \)). This is consistent with the law of laminar flow, where the velocity is highest along the central axis of the artery.

Step 4: Create table of values

Calculate the velocity \( v(r) \) for each specified \( r \) value and construct a table:- For \( r = 0 \):\[v(0) = 18,500 \times 0.25 = 4,625 \, \text{cm/s}\]- For \( r = 0.1 \):\[v(0.1) = 4,440 \, \text{cm/s}\]- For \( r = 0.2 \):\[v(0.2) = 18,500 \times (0.25 - 0.04) = 18,500 \times 0.21 = 3,885 \, \text{cm/s}\]- For \( r = 0.3 \):\[v(0.3) = 18,500 \times (0.25 - 0.09) = 18,500 \times 0.16 = 2,960 \, \text{cm/s}\]- For \( r = 0.4 \):\[v(0.4) = 1,665 \, \text{cm/s}\]- For \( r = 0.5 \):\[v(0.5) = 18,500 \times (0.25 - 0.25) = 18,500 \times 0 = 0 \, \text{cm/s}\]The table summarizes these results:\[\begin{array}{|c|c|}\hliner & v(r) \ \hline0 & 4,625 \, \text{cm/s} \0.1 & 4,440 \, \text{cm/s} \0.2 & 3,885 \, \text{cm/s} \0.3 & 2,960 \, \text{cm/s} \0.4 & 1,665 \, \text{cm/s} \0.5 & 0 \, \text{cm/s} \\hline\end{array}\]

Key Concepts

Blood Velocity in Arteries

Blood velocity in arteries is a captivating aspect of biological fluid dynamics. When analyzing the movement of blood through arteries, it's essential to understand that velocity varies based on the radial distance from the artery's center.
The blood velocity is highest at the central axis, decreasing steadily as you move towards the artery walls. This phenomenon is explained by the law of laminar flow, where blood layers move in parallel paths without mixing.

• The central axis exhibits the maximum velocity due to less friction with artery walls.
• Blood near the wall faces more resistance and hence, shows reduced velocity.
• This gradient creates a parabolic velocity profile across different radial positions.

Understanding this concept helps comprehend basic cardiovascular physiology and blood flow in arteries, fundamental for grasping more complex biological and medical principles.

Mathematical Functions in Biology

Mathematical functions play a crucial role in modeling biological processes like blood flow in arteries. The formula provided in the exercise, \( v(r) = 18,500(0.25 - r^2) \), elegantly captures how velocity changes with distance from the artery's center.
Such functions:

• Help in predicting velocity at any given point by simply substituting the radius in the function.
• Enable a mathematical representation of real-life biological phenomena.
• Facilitate deeper analysis and understanding of complex biological systems through calculus and derivatives.

In biology, employing functions like this enables researchers, doctors, and students to apply mathematical rigor to physiological concepts, thus enhancing overall learning and diagnostic capabilities.

Central Axis Velocity

The concept of central axis velocity is pivotal in understanding blood flow dynamics within an artery. It refers to the maximum speed of blood flow that occurs precisely down the center of a blood vessel.
The exercise elaborates this with the formulated expression, where substituting \( r = 0 \) demonstrates the highest velocity. Here,

• When \( r = 0 \), \( v(0) = 18,500 \times 0.25 = 4,625 \, \text{cm/s} \).
• The central point boasts maximum flow due to minimal friction with the artery's walls.
• This maximum velocity is central to understanding vessel efficiency and function.

Grasping central axis velocity offers insights into cardiovascular efficiency and can guide therapeutic strategies in medical sciences.