Problem 63
Question
Unclogging arteries The formula \(V=k r^{4},\) discovered by the physiologist Jean Poiseuille (1797-1869), allows us to predict how much the radius of a partially clogged artery has to be expanded in order to restore normal blood flow. The formula says that the volume \(V\) of blood flowing through the artery in a unit of time at a fixed pressure is a constant \(k\) times the radius of the artery to the fourth power. How will a \(10 \%\) increase in \(r\) affect \(V ?\)
Step-by-Step Solution
Verified Answer
The volume increases by approximately 46.41%.
1Step 1: Understanding the Formula
The formula given is \( V = k r^4 \), where \( V \) represents the volume of blood flow, \( k \) is a constant, and \( r \) is the radius of the artery. Our task is to calculate how a 10% increase in \( r \) affects \( V \).
2Step 2: Calculating the New Radius
If the original radius is \( r \), a 10% increase means the new radius \( r' \) is \( r' = r + 0.1r = 1.1r \).
3Step 3: Substitute New Radius into Formula
Substitute \( r' = 1.1r \) into the original formula to find the new volume: \( V' = k (1.1r)^4 \).
4Step 4: Simplifying the Expression
Calculate \( (1.1r)^4 \): \((1.1)^4 = 1.4641\), so \( V' = k(1.4641)r^4 \).
5Step 5: Comparing Old and New Volume
The original volume is \( V = kr^4 \). The new volume is \( V' = 1.4641kr^4 \). Therefore, \( V' = 1.4641V \).
6Step 6: Conclusion
A 10% increase in radius results in a volume \( V' \) that is approximately 46.41% greater than the original volume \( V \).
Key Concepts
Poiseuille's LawBlood FlowVolume CalculationRadius Increase
Poiseuille's Law
Poiseuille's Law is a fundamental principle in fluid mechanics that is specifically applicable to the flow of incompressible fluids, like blood, through a narrow pipe or tube. Developed by Jean Léonard Marie Poiseuille, this law provides a clear relationship between the flow rate, the pressure difference, and the dimensions of the pipe, particularly its radius. In this context, the flow rate is dictated by the equation \( V = k r^4 \). Here, \( V \) stands for the volume of fluid passing through in a given time, \( r \) is the radius of the tube, and \( k \) is a constant that incorporates factors like the fluid's viscosity and the pressure gradient across the ends of the tube.
This law famously emphasizes how sensitive the flow rate is to changes in the tube’s radius—the relationship is to the fourth power. This means that even a small increase in the radius can significantly amplify the flow of fluid.
This law famously emphasizes how sensitive the flow rate is to changes in the tube’s radius—the relationship is to the fourth power. This means that even a small increase in the radius can significantly amplify the flow of fluid.
Blood Flow
Understanding blood flow is crucial as it carries essential oxygen and nutrients to cells throughout the body. Blood flow in arteries can be affected by various factors including the vessel's radius. According to Poiseuille's Law, the flow rate of blood is highly dependent on the fourth power of the radius of the vessel. This implies that slight changes in the radius can lead to dramatic changes in blood flow.
In medical scenarios, especially with arteriosclerosis (narrowing of arteries), it's important to provide optimal conditions for blood flow. By unclogging or widening an artery, we can restore or significantly improve blood circulation in the body. Hence, understanding and applying this principle has tremendous implications for medical health and interventions.
In medical scenarios, especially with arteriosclerosis (narrowing of arteries), it's important to provide optimal conditions for blood flow. By unclogging or widening an artery, we can restore or significantly improve blood circulation in the body. Hence, understanding and applying this principle has tremendous implications for medical health and interventions.
Volume Calculation
Volume calculation in the context of fluid mechanics involves determining how much of a fluid passes through a cross-sectional area over time. With Poiseuille's framework, the equation \( V = k r^4 \) simplifies this process. Here, volume \( V \) is calculated by considering the radius \( r \) of the pipe and adjusting for various constants that affect flow, such as fluid viscosity.
A 10% increase in radius is triumphed in the calculation: substitute the new radius \( r' = 1.1r \) into the formula to see the impact. After computing \( (1.1)^4 \), we arrive at approximately 1.4641. This means the new volume becomes \( V' = 1.4641k r^4 \), which represents a substantial 46.41% increase in volume from the original. This step underscores the powerful impact that size adjustments have on volume flow.
A 10% increase in radius is triumphed in the calculation: substitute the new radius \( r' = 1.1r \) into the formula to see the impact. After computing \( (1.1)^4 \), we arrive at approximately 1.4641. This means the new volume becomes \( V' = 1.4641k r^4 \), which represents a substantial 46.41% increase in volume from the original. This step underscores the powerful impact that size adjustments have on volume flow.
Radius Increase
An increase in the radius of a vessel, like an artery, can profoundly affect the flow rate of blood due to the relationship outlined in Poiseuille's Law. Given the formula \( V = k r^4 \), even small changes in the radius have amplified effects on volume flow.
When calculating a 10% increase in radius, indicated by \( r' = 1.1r \), the resulting effect on volume is large—nearly a 46.41% increase in flow rate. The mathematical transformation \( r^4 \rightarrow (1.1r)^4 \), which equals \( 1.4641r^4 \), starkly illustrates the sensitivity of flow volume to radius changes.
Understanding this exponential relationship can assist in making informed decisions in medical treatments and interventions where precise control of flow rates is required, such as in angioplasty procedures.
When calculating a 10% increase in radius, indicated by \( r' = 1.1r \), the resulting effect on volume is large—nearly a 46.41% increase in flow rate. The mathematical transformation \( r^4 \rightarrow (1.1r)^4 \), which equals \( 1.4641r^4 \), starkly illustrates the sensitivity of flow volume to radius changes.
Understanding this exponential relationship can assist in making informed decisions in medical treatments and interventions where precise control of flow rates is required, such as in angioplasty procedures.
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