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55) A typical arteriole has a diameter of 0.080 mm and carries blood at the rate of 9.6×10^−5 cm3/s. (a) What is the speed of the blood in an arteriole? (b) Suppose an arteriole branches into 8800 capillaries, each with a diameter of 6.0×10^−6 m. What is the blood speed in the capillaries? (The low speed in capillaries is beneficial; it promotes the diffusion of materials to and from the blood.)

Sagot :

ANSWER:

a) 1.91 cm/s

b) 0.039 cm/s

STEP-BY-STEP EXPLANATION:

a)

Here, since we need the velocity in cm/s, we assume that a small volume of blood passes through the section of the arteriole in a given time.

So, we have to divide the blood flow by the area of the arteriole section, therefore:

[tex]A=\pi\cdot\mleft(\frac{d}{2}\mright)^2[/tex]

d = 0.08 mm = 0.008 cm

Replacing:

[tex]A=3.14\cdot\mleft(\frac{0.008}{2}\mright)^2=0.00005024=5.024\cdot10^{-5}cm^2[/tex]

We calculate the speed by dividing the rate by the previous calculated area, like this:

[tex]\begin{gathered} v=\frac{q}{A} \\ \text{Replacing} \\ v=\frac{9.6\cdot10^{-5}}{5.024\cdot10^{-5}} \\ v=1.91\text{ cm/s} \end{gathered}[/tex]

b)

First we find the area of the section of the capillaries:

[tex]\begin{gathered} d=6\cdot10^{-6}m=6\cdot10^{-4}cm \\ R\text{eplacing} \\ A=3.14\cdot\mleft(\frac{6\cdot10^{-4}}{2}\mright)^2 \\ A=0.0000002826=2.826\cdot10^{-7}cm^2 \end{gathered}[/tex]

Now we have to remember that the flow is dividid in equal parts, so the volume by seconds is:

[tex]\begin{gathered} q=\frac{9.6\cdot10^{-5}}{8800} \\ q=1.09\cdot10^{-8}\frac{cm^3}{s} \end{gathered}[/tex]

So the speed in this case is:

[tex]\begin{gathered} v=\frac{q}{A} \\ v=\frac{1.09\cdot10^{-8}}{2.826\cdot10^{-7}} \\ v=0.039\text{ cm/s} \end{gathered}[/tex]

This latter speed is less than in the main arteriole.