A typical arteriole has a diameter of 0.080 mm and carries blood at the rate of 9.6×10−5cm3/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 6.0×10−6m.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.
a. Well, if you're looking for speed, you won't find any in an arteriole traffic jam! But in all seriousness, to calculate the speed of the blood in an arteriole, we need to use the equation:
Speed = Flow rate / Cross-sectional area
First, we need to convert the diameter of the arteriole to meters:
Diameter = 0.080 mm = 0.080 × 10^(-3) m
Next, let's calculate the cross-sectional area of the arteriole:
Area = π × (radius)^2
= π × (0.080 × 10^(-3) m / 2)^2
The flow rate is already given as 9.6 × 10^(-5) cm^3/s, but we need to convert it to m^3/s:
Flow rate = 9.6 × 10^(-5) cm^3/s = 9.6 × 10^(-8) m^3/s
Now, we can calculate the speed:
Speed = 9.6 × 10^(-8) m^3/s / (π × (0.080 × 10^(-3) m / 2)^2)
b. Oh boy, branching out into 8800 capillaries? That's a big family reunion! To calculate the blood speed in the capillaries, we can use the same equation:
Speed = Flow rate / Cross-sectional area
First, let's calculate the cross-sectional area of one capillary:
Area = π × (radius)^2
= π × (6.0 × 10^(-6) m / 2)^2
The flow rate remains the same as before, but now we have 8800 capillaries, so our total flow rate is:
Total flow rate = 9.6 × 10^(-8) m^3/s × 8800
Now, we can calculate the speed in the capillaries:
Speed = (9.6 × 10^(-8) m^3/s × 8800) / (π × (6.0 × 10^(-6) m / 2)^2)
Remember, in the world of capillaries, slow and steady wins the race!
a. To find the speed of blood in an arteriole, we can use the formula:
Speed = Flow rate / Cross-sectional area
First, we need to convert the given diameter from millimeters to meters:
Diameter = 0.080 mm = 0.080 × 10^(-3) m = 8.0 × 10^(-5) m
Next, we need to calculate the cross-sectional area of the arteriole:
Area = π × (radius)^2
Radius = Diameter / 2 = 8.0 × 10^(-5) m / 2 = 4.0 × 10^(-5) m
Area = π × (4.0 × 10^(-5))^2 = 5.0 × 10^(-9) m^2
Now we can calculate the speed of the blood:
Speed = 9.6 × 10^(-5) cm^3/s / (5.0 × 10^(-9) m^2)
= (9.6 × 10^(-5) cm^3/s) / (5.0 × 10^(-9) m^2)
To convert cm^3 to m^3, we multiply by (0.01 m / 1 cm)^3:
Speed = (9.6 × 10^(-5) cm^3/s) / (5.0 × 10^(-9) m^2) × (0.01 m / 1 cm)^3
Simplifying, we get:
Speed = 1.92 × 10^(-3) m/s
Therefore, the speed of the blood in an arteriole is 1.92 × 10^(-3) m/s.
b. To find the blood speed in the capillaries, we can use the principle of continuity, which states that the product of area and speed must remain constant along a fluid's flow.
According to the principle of continuity:
A1 × V1 = A2 × V2
Where:
A1 = Cross-sectional area of the arteriole
V1 = Speed of the blood in the arteriole
A2 = Cross-sectional area of the capillaries
V2 = Speed of the blood in the capillaries
We already calculated the speed in the arteriole as 1.92 × 10^(-3) m/s in part a. Now, we need to determine the cross-sectional area of the capillaries.
Radius = Diameter / 2 = 6.0 × 10^(-6) m / 2 = 3.0 × 10^(-6) m
Area = π × (3.0 × 10^(-6))^2 = 2.8 × 10^(-11) m^2
Now, we can find the speed in the capillaries:
A1 × V1 = A2 × V2
(5.0 × 10^(-9) m^2) × (1.92 × 10^(-3) m/s) = (2.8 × 10^(-11) m^2) × V2
V2 = (5.0 × 10^(-9) m^2) × (1.92 × 10^(-3) m/s) / (2.8 × 10^(-11) m^2)
Simplifying, we get:
V2 ≈ 3.43 × 10^(-4) m/s
Therefore, the blood speed in the capillaries is approximately 3.43 × 10^(-4) m/s.
To find the blood speed in the arteriole and then in the capillaries, we can use the equation for flow rate (Q) and the cross-sectional area (A) of the blood vessel. The equation is as follows:
Q = A * v
where Q is the flow rate, A is the cross-sectional area, and v is the velocity of the blood.
a. To find the speed of the blood in the arteriole, we need to calculate the velocity (v). We are given the diameter (d) of the arteriole, which we can use to calculate the cross-sectional area (A).
The formula to calculate the cross-sectional area of a circle is:
A = π * r^2
where r is the radius of the circle.
Given the diameter of the arteriole is 0.080 mm, we can calculate the radius:
r = d/2 = 0.080 mm / 2 = 0.040 mm = 0.040 * 10^(-3) m
Now we can calculate the cross-sectional area (A):
A = π * (0.040 * 10^(-3))^2
Using the given flow rate (9.6×10^(-5) cm^3/s), we can substitute the values in the flow rate equation:
9.6 * 10^(-5) cm^3/s = A * v
Now solve for v:
v = (9.6 * 10^(-5) cm^3/s) / A
Substituting the value of A, we can calculate the velocity (v) in cm/s.
b. To find the blood speed in the capillaries, we will use the principle of conservation of mass. As the arteriole branches into multiple capillaries, the flow rate remains the same throughout the vessel system.
First, we need to calculate the total cross-sectional area of all the capillaries. Since we are given the diameter of one capillary, we can calculate the radius and then the cross-sectional area. Then multiply it by the number of capillaries (8800) to get the total cross-sectional area (A_capillaries).
Now that we have the total cross-sectional area (A_capillaries), we can use the flow rate equation (Q = A * v) to find the velocity (v) in the capillaries. Since the flow rate (Q) remains the same, we can rearrange the equation to solve for v:
v = Q / A_capillaries
Substituting the value of Q and A_capillaries, we can calculate the velocity (v) in cm/s.
a. speed*area=rate
speed= 9.6e-5/(PI*.004^2) cm/sec
b. same formula, except rate is not 1/8800. Notice diameter is in meters, suggest change to cm first.