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Chapter 3: Problem 11

The arterioles (small arteries) leading to an organ, constrict in order to decrease flow to the organ. To shut down an organ, blood flow is reduced naturally to \(1.00 \%\) of its original value. By what factor did the radii of the arterioles constrict? Penguins do this when they stand on ice to reduce the blood flow to their feet.

Short Answer

Expert verified
The radii of the arterioles constrict to one-tenth (0.1 times) of their original size.

Step by step solution

01

Understand the relationship between flow rate, area, and radius

The flow rate of blood through a vessel is proportional to the cross-sectional area of the vessel, which is determined by the square of the radius by the equation for the area of a circle, A = \( \(\pi r^2\) \). Therefore, if we want to reduce the blood flow to 1% of its original value, the cross-sectional area must also reduce to 1% of its original value due to the direct proportionality.
02

Establish the relationship between the original and constricted radii

Let's denote the original radius of the arterioles as \( r_{o} \) and the constricted radius as \( r_{c} \). We have the relationship between the original area \( A_{o} \) and the constricted area \( A_{c} \) such that \( A_{c} = 0.01A_{o} \). Since area is proportional to radius squared, we can write \( (r_{c})^2 = 0.01(r_{o})^2 \).
03

Solve for the constricted radius in terms of the original radius

Taking the square root of both sides of the equation, we get \( r_{c} = \sqrt{0.01}(r_{o}) \). Simplifying the square root of 0.01 gives us \( r_{c} = 0.1r_{o} \).
04

Determine the ratio of constricted radii to original radii

The factor by which the radii of the arterioles constrict is therefore \( r_{c} / r_{o} = 0.1 \), meaning the radii constrict to one-tenth of their original size.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Blood Flow Regulation
The human body is equipped with an intricate system to regulate blood flow to different organs based on various physiological needs. Regulation is critical for maintaining homeostasis, ensuring that organs receive the right amount of blood according to their immediate requirements. This is particularly important in response to environmental changes, such as exposure to cold temperatures.

One way the body manages blood flow is through the constriction and dilation of blood vessels. Arterioles, which are small branches of arteries leading into the capillary beds of tissues, play a key role in this regulatory process. They are muscular and can constrict or dilate to change the volume of blood flowing to specific areas of the body.
Arteriole Radius Constriction
In conditions like cold environments, arterioles undergo a process known as vasoconstriction, where they narrow in diameter. This radius constriction is a critical adaptation for reducing heat loss and preserving core body temperature. The arterioles achieve this by contracting the smooth muscle in their walls, thereby reducing the diameter of the vessel lumen.

Arteriole radius constriction can be explained using the principle of Poiseuille's law, which relates flow rate through a vessel to the radius of the vessel. A smaller radius leads to increased resistance to blood flow, which reduces the flow rate. For instance, when penguins stand on ice, they constrict their blood vessels to reduce blood flow to their feet, thereby minimizing heat loss.
Cross-Sectional Area and Flow Rate
The relationship between the cross-sectional area of a blood vessel and the flow rate of blood through it is given by the equation for volumetric flow rate, which states that the flow rate is proportional to the cross-sectional area of the vessel. This area is a function of the radius of the arterioles, represented by the equation for the area of a circle, A = \( \pi r^2 \).

When arterioles constrict, their radius decreases, leading to a dramatic reduction in their cross-sectional area and consequently, the flow rate. According to the exercise problem, when the radius reduces, the cross-sectional area is just 1% of its original value, leading to a flow rate that is also 1% of the original, reflecting a substantial decrease in blood delivered to the specific organ or tissue.

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Most popular questions from this chapter

Bryan Allen pedaled a human-powered aircraft across the English Channel from the cliffs of Dover to Cap Gris-Nez on June \(12,1979 .\) (a) He flew for 169 min at an average velocity of \(3.53 \mathrm{m} / \mathrm{s}\) in a direction \(45^{\circ}\) south of east. What was his total displacement? (b) Allen encountered a headwind averaging \(2.00 \mathrm{m} / \mathrm{s}\) almost precisely in the opposite direction of his motion relative to the Earth. What was his average velocity relative to the air? (c) What was his total displacement relative to the air mass?

The velocity of the wind relative to the water is crucial to sailboats. Suppose a sailboat is in an ocean current that has a velocity of \(2.20 \mathrm{m} / \mathrm{s}\) in a direction \(30.0^{\circ}\) east of north relative to the Earth. It encounters a wind that has a velocity of \(4.50 \mathrm{m} / \mathrm{s}\) in a direction of \(50.0^{\circ}\) south of west relative to the Earth. What is the velocity of the wind relative to the water?

You drive \(7.50 \mathrm{km}\) in a straight line in a direction \(15^{\circ}\) east of north. (a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.) (b) Show that you still arrive at the same point if the east and north legs are reversed in order.

Solve the following problem using analytical techniques: Suppose you walk \(18.0 \mathrm{m}\) straight west and then \(25.0 \mathrm{m}\) straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements \(\mathbf{A}\) and \(\mathbf{B},\) as in Figure \(3.58,\) then this problem asks you to find their sum \(\mathbf{R}=\mathbf{A}+\mathbf{B} .)\)

Calculate the Reynolds numbers for the flow of water through (a) a nozzle with a radius of \(0.250 \mathrm{~cm}\) and (b) a garden hose with a radius of \(0.900 \mathrm{~cm}\), when the nozzle is attached to the hose. The flow rate through hose and nozzle is \(0.500\) L/s. Can the flow in either possibly be laminar?
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