A)Blood flows in an aorta of radius 4.50mm at 27.5cm/s. Calculate the volume flow rate.
B)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 0.85mm/s, calculate the total cross-sectional area of the capillaries.
A) To calculate the volume flow rate, we can use the equation:
Volume Flow Rate = Cross-sectional Area * Velocity
Given:
Radius of the aorta (r) = 4.50 mm = 0.45 cm
Velocity (v) = 27.5 cm/s
First, we need to calculate the cross-sectional area of the aorta:
Cross-sectional Area = π * r^2
Cross-sectional Area = π * (0.45 cm)^2
Now, substitute the cross-sectional area and velocity values into the formula:
Volume Flow Rate = (π * (0.45 cm)^2) * 27.5 cm/s
Using the value of π (approximately 3.14159), we can calculate the volume flow rate.
B) To calculate the total cross-sectional area of the capillaries, we need to use the concept of continuity of flow. According to the principle of continuity, the volume flow rate remains constant throughout any given system of connected vessels.
Given:
Velocity through the capillaries (v) = 0.85 mm/s
Using the same volume flow rate from part A, we can set up the equation:
Volume Flow Rate (aorta) = Volume Flow Rate (capillaries)
Cross-sectional Area (aorta) * Velocity (aorta) = Cross-sectional Area (capillaries) * Velocity (capillaries)
Rearranging the equation, we can solve for the total cross-sectional area of the capillaries:
Cross-sectional Area (capillaries) = (Cross-sectional Area (aorta) * Velocity (aorta)) / Velocity (capillaries)
Substitute the values we know:
Cross-sectional Area (capillaries) = ((π * (0.45 cm)^2) * 27.5 cm/s) / 0.85 mm/s
Using the appropriate unit conversions, such as converting mm to cm, we can calculate the total cross-sectional area of the capillaries.