Blood flows in an aorta of radius 4.62mm at 35.0cm/s. Calculate the volume flow rate.
Although the cross-sectional area of a capillary is much smaller than that of the aorta, there are many capillaries, so their total cross-sectional area is much larger. If all the blood from the aorta flows into the capillaries and the speed of flow through the capillaries is 1.10mm/s, calculate the total cross-sectional area of the capillaries.
To calculate the volume flow rate in the aorta, we can use the equation:
Q = A * v
where Q is the volume flow rate, A is the cross-sectional area of the aorta, and v is the velocity of blood flow.
Given:
Radius of the aorta (r) = 4.62 mm = 0.462 cm
Velocity of blood flow in the aorta (v) = 35.0 cm/s
First, we can calculate the cross-sectional area of the aorta (A) using the formula:
A = π * r^2
A = π * (0.462 cm)^2
A ≈ 0.670 cm^2
Now, we can substitute the values into the equation for volume flow rate:
Q = 0.670 cm^2 * 35.0 cm/s
Q ≈ 23.45 cm^3/s
Therefore, the volume flow rate in the aorta is approximately 23.45 cm^3/s.
Now, to calculate the total cross-sectional area of the capillaries, we can use the equation:
Q = A * v
where Q is the volume flow rate in the aorta, A is the total cross-sectional area of the capillaries, and v is the velocity of blood flow through the capillaries.
Given:
Volume flow rate in the aorta (Q) = 23.45 cm^3/s
Velocity of blood flow through the capillaries (v) = 1.10 mm/s
We need to convert the volume flow rate from cm^3/s to mm^3/s because the velocity unit is in mm/s.
1 cm^3 = 1000 mm^3
Q = 23.45 cm^3/s * 1000 mm^3/cm^3
Q = 23450 mm^3/s
Now, we can substitute the values into the equation for volume flow rate:
23450 mm^3/s = A * 1.10 mm/s
Divide both sides by 1.10 mm/s to solve for A:
A = 23450 mm^3/s / 1.10 mm/s
A ≈ 21318.18 mm^2
Therefore, the total cross-sectional area of the capillaries is approximately 21318.18 mm^2.