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Chapter 9: Problem 86

The average speed of blood in the aorta is \(0.3 \mathrm{m} / \mathrm{s}\) and the radius of the aorta is \(1 \mathrm{cm} .\) There are about \(2 \times 10^{9}\) capillaries with an average radius of \(6 \mu \mathrm{m}\). What is the approximate average speed of the blood flow in the capillaries?

Short Answer

Expert verified
Solution: 1. Calculate the volume flow rate in the aorta: Q_aorta = 0.3 m/s × π × (0.01 m)² 2. Calculate the total cross-sectional area of capillaries: A_cap = π × (6 x 10^(-6) m)² A_total_cap = 2 x 10^9 × A_cap 3. Determine the volume flow rate in the capillaries: Q_capillaries = Q_aorta 4. Calculate the average speed of blood flow in the capillaries: v_cap = Q_capillaries / A_total_cap Using the calculated values from steps 1 and 2, find the approximate average speed of blood flow in the capillaries.

Step by step solution

01

Calculate the volume flow rate in the aorta.

The volume flow rate can be calculated using the formula: Volume flow rate (Q) = Velocity (v) × Cross-sectional area (A) For the aorta, we have the velocity v = 0.3 m/s and the radius r = 1 cm = 0.01 m. The cross-sectional area of the aorta is given by A=πr². So let's calculate the volume flow rate in the aorta: Q_aorta = v × A_aorta = 0.3 m/s × π × (0.01 m)²
02

Calculate the total cross-sectional area of capillaries.

We are given the average radius of each capillary as r_cap = 6 µm = 6 x 10^(-6) m and the total number of capillaries as N = 2 x 10^9. Then, we can calculate the cross-sectional area of one capillary, A_cap = π × (r_cap)², and the total cross-sectional area of all capillaries, A_total_cap = N × A_cap.
03

Calculate the volume flow rate in the capillaries.

Since the volume flow rate in the aorta and in the capillaries should be equal, we have: Q_capillaries = Q_aorta
04

Calculate the average speed of blood flow in the capillaries.

Now, we can use the formula for volume flow rate in the capillaries and solve for the average speed (v_cap) of the blood flow in the capillaries: Q_capillaries = v_cap × A_total_cap So, v_cap = Q_capillaries / A_total_cap Plugging in the values obtained from step 1 and step 2, we can find the approximate average speed of the blood flow in the capillaries.

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