(a) Calculate the average blood velocity in the major arties given that the aorta has a radius of 1.0 cm, the blood velocity is 30 cm/s in the aorta, and the total cross-sectional area of the major arties is 20 cm2 (b) What is the total flow rate? (c) On the assumption that all the blood in the circulatory system goes through the capillaries, and given that the average velocity of the blood in the capillaries is 0.03 cm/s, what is the total cross-sectional area of the capillaries?
To calculate the average blood velocity in the major arteries, you can use the principle of continuity, which states that the product of cross-sectional area and velocity is constant throughout a closed system.
(a) Average blood velocity in major arteries:
1. Start by calculating the cross-sectional area of the aorta and the major arteries combined. Since the total cross-sectional area of the major arteries is given as 20 cm^2, subtract the cross-sectional area of the aorta to find the cross-sectional area of the major arteries alone. Let's assume the cross-sectional area of the aorta is 3.14 cm^2 (radius of 1.0 cm).
Total cross-sectional area of major arteries = 20 cm^2 - 3.14 cm^2
= 16.86 cm^2
2. Now, use the principle of continuity to find the average blood velocity in the major arteries.
Average blood velocity = Total blood flow rate / Total cross-sectional area of major arteries
The total blood flow rate will be the same throughout the major arteries.
Given that the blood velocity in the aorta is 30 cm/s, the equation becomes:
30 cm/s = Total blood flow rate / 16.86 cm^2
Rearranging the equation, we find:
Total blood flow rate = 30 cm/s * 16.86 cm^2
Total blood flow rate = 505.8 cm^3/s
3. Finally, to calculate the average blood velocity in the major arteries, divide the total blood flow rate by the total cross-sectional area:
Average blood velocity = Total blood flow rate / Total cross-sectional area of major arteries
Average blood velocity = 505.8 cm^3/s / 16.86 cm^2
Average blood velocity ≈ 30 cm/s
(b) Total flow rate:
The total flow rate is already calculated in part (a) as approximately 505.8 cm^3/s.
(c) Total cross-sectional area of the capillaries:
To find the total cross-sectional area of the capillaries, you can use the same principle of continuity.
Total blood flow rate = Average blood velocity in capillaries * Total cross-sectional area of capillaries
Given the average blood velocity in capillaries as 0.03 cm/s, you can rewrite the equation:
505.8 cm^3/s = 0.03 cm/s * Total cross-sectional area of capillaries
Rearranging the equation, you get:
Total cross-sectional area of capillaries = 505.8 cm^3/s / 0.03 cm/s
Total cross-sectional area of capillaries ≈ 16,860 cm^2
(a) To calculate the average blood velocity in the major arteries, we need to use the principle of conservation of flow, which states that the flow rate is constant throughout a closed system.
First, let's calculate the flow rate in the aorta:
Flow rate in the aorta = Blood velocity in the aorta * Cross-sectional area of the aorta
Flow rate in the aorta = 30 cm/s * π * (1.0 cm)^2
Flow rate in the aorta = 30π cm^3/s
Next, we can calculate the flow rate in the major arteries:
Flow rate in the arteries = Total flow rate - Flow rate in the aorta
Flow rate in the arteries = Total flow rate - 30π cm^3/s
To find the average blood velocity in the arteries, we need to divide the flow rate by the total cross-sectional area of the major arteries:
Average blood velocity in the arteries = Flow rate in the arteries / Total cross-sectional area of the major arteries
Average blood velocity in the arteries = (Total flow rate - 30π cm^3/s) / 20 cm^2
(b) The total flow rate is the same throughout the circulatory system. Therefore, the total flow rate is equal to the flow rate in the aorta.
Total flow rate = Flow rate in the aorta = 30π cm^3/s
(c) To find the total cross-sectional area of the capillaries, we can use the principle of conservation of flow again:
Total flow rate = Average blood velocity in the capillaries * Total cross-sectional area of the capillaries
We are given the average blood velocity in the capillaries (0.03 cm/s). Re-arranging the equation above:
Total cross-sectional area of the capillaries = Total flow rate / Average blood velocity in the capillaries
Total cross-sectional area of the capillaries = 30π cm^3/s / 0.03 cm/s