/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 81 The approximate diameter of the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 81

The approximate diameter of the aorta is \(0.50 \mathrm{~cm}\); that of a capillary is \(10 \mu \mathrm{m}\). The approximate average blood flow speed is \(1.0 \mathrm{~m} / \mathrm{s}\) in the aorta and \(1.0 \mathrm{~cm} / \mathrm{s}\) in the capillaries. If all the blood in the aorta eventually flows through the capillaries, estimate the number of capillaries in the circulatory system.

Short Answer

Expert verified
The estimated number of capillaries in the circulatory system is given by the formula \( n = \frac{R_{aorta}}{A_{capillary} \times V_{capillary}} \). Substitute the known values and calculate n. Be sure to use appropriate units before calculating.

Step by step solution

01

Calculate the Area of the Aorta and Capillaries

First, calculate the cross-sectional area of both the aorta and the capillary using the formula for the area of a circle: \( A = \pi r^{2} \). Given the diameters, the radius is half the diameter. So, for the Aorta \( A_{aorta} = \pi (0.25 cm)^{2} \) and for the Capillary \( A_{capillary} = \pi (5 \mu m)^{2} \). But remember to convert the radii to appropriate units before calculating. Here, convert cm to m and 碌m to m.
02

Calculate the Flow Rate in the Aorta

The flow rate is calculated by multiplying the cross-sectional area by the speed of the flow. In this case, the flow rate in the aorta \( R_{aorta} = A_{aorta} \times V_{aorta} \), where \( V_{aorta} = 1.0 m/s \) is the speed of the blood flow in the aorta. Remember to use SI units before multiplying.
03

Calculate the Number of Capillaries

Using the principle of continuity, the total flow rate in the aorta is equal to the total flow rate in the capillaries combined. Hence, \( R_{aorta} = n \times A_{capillary} \times V_{capillary} \), where n is the total number of capillaries we want to find and \( V_{capillary} = 1.0 cm/s \). Simplify this equation to solve for n: \( n = \frac{R_{aorta}}{A_{capillary} \times V_{capillary}} \). Substitute the known values and calculate n to get the estimated number of capillaries in the circulatory system.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Aorta
The aorta is the largest artery in the human body, playing a crucial role in the circulatory system. It is responsible for carrying oxygen-rich blood from the heart to the rest of the body.
The diameter of the aorta is approximately 0.50 cm, which translates into a radius of 0.25 cm.
  • This large size allows it to handle high pressures as blood is ejected from the heart.
  • The wide diameter reduces resistance, ensuring efficient blood flow.
The aorta's substantial diameter compared to other blood vessels helps distribute blood quickly, making it an essential component in fluid dynamics studies within the human body.
Capillaries
Capillaries are the smallest blood vessels in the body, with a diameter of about 10 micrometers (\(10 \mu \text{m} \)). These tiny vessels are where the exchange of nutrients, gases, and waste products occurs between blood and tissues.
Despite their small size, capillaries are numerous, forming an extensive network throughout the body.
  • This massive network ensures sufficient surface area for exchange processes.
  • Their thin walls facilitate the rapid diffusion of substances.
Since they have a much smaller diameter compared to the aorta, the blood flow speed is also reduced, which allows more time for the exchange of substances.
Blood Flow Rate
Blood flow rate in the circulatory system is a key concept in fluid dynamics, indicating the volume of blood passing through a vessel per unit time. It is influenced by the cross-sectional area of the vessel and the velocity of blood flow.
In the aorta, the blood flow speed is approximately 1.0 m/s, while in the capillaries, it drops to 1.0 cm/s.
  • The flow rate in any section of the circulatory system can be calculated using the formula: \( R = A \times V \), where \( R \) is the flow rate, \( A \) is the cross-sectional area, and \( V \) is the flow speed.
  • A higher speed in the aorta is due to its larger diameter, facilitating rapid transport of blood.
  • Lower speed in the capillaries ensures an adequate time for exchange processes.
Understanding blood flow rate is essential for calculating the number of capillaries that can handle the blood circulating from the aorta.
Continuity Equation
The continuity equation is a fundamental principle in fluid dynamics, stating that the flow rate must remain constant from one cross-section of a vessel to another.This principle is applied in the circulatory system to ensure the same volume of blood that enters the capillaries via the aorta exits without loss.
  • For the aorta and capillaries, it is expressed as \( R_{\text{aorta}} = n \times A_{\text{capillary}} \times V_{\text{capillary}} \).
  • This equation helps determine the number \( n \) of capillaries by rearranging it to solve for \( n \).
  • The continuity equation is crucial for analyzing how changes in vessel size and flow velocity affect the overall blood distribution.
By utilizing the continuity equation, one can effectively estimate complex biological structures like the extensive capillary network.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A collapsible plastic bag (Fig. P9.23) contains a glucose solution. If the average gauge pressure in the vein is \(1.33 \times 10^{3} \mathrm{~Pa}\), what must be the minimum height \(h\) of the bag in order to infuse glucose into the vein? Assume the specific gravity of the solution is \(1.02\).

A cube of wood having an edge dimension of \(20.0 \mathrm{~cm}\) and a density of \(650 \mathrm{~kg} / \mathrm{m}^{3}\) floats on water. (a) What is the distance from the horizontal top surface of the cube to the water level? (b) What mass of lead should be placed on the cube so that the top of the cube will be just level with the water surface?

A straight horizontal pipe with a diameter of \(1.0 \mathrm{~cm}\) and a length of \(50 \mathrm{~m}\) carries oil with a coefficient of viscosity of \(0.12 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\). At the output of the pipe, the flow rate is \(8.6 \times 10^{-5} \mathrm{~m}^{3} / \mathrm{s}\) and the pressure is \(1.0 \mathrm{~atm}\). Find the gauge pressure at the pipe input.

Whole blood has a surface tension of \(0.058 \mathrm{~N} / \mathrm{m}\) and a density of \(1050 \mathrm{~kg} / \mathrm{m}^{3}\). To what height can whole blood rise in a capillary blood vessel that has a radius of \(2.0 \times 10^{-6} \mathrm{~m}\) if the contact angle is zero?

What radius needle should be used to inject a volume of \(500 \mathrm{~cm}^{3}\) of a solution into a patient in \(30 \mathrm{~min}\) ? Assume the length of the needle is \(2.5 \mathrm{~cm}\) and the solution is elevated \(1.0 \mathrm{~m}\) above the point of injection. Further, assume the viscosity and density of the solution are those of pure water, and that the pressure inside the vein is atmospheric.
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Energy Physics

Read Explanation

Nuclear Physics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Engineering Physics

Read Explanation

Solid State Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.