Blood flows in an aorta of radius 4.12mm 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:
Volume flow rate = Area × Velocity
First, let's calculate the area of the aorta using the given radius:
Area = π × radius^2
Area = π × (4.12mm)^2
Next, convert the radius to meters to ensure consistent units:
Area = π × (0.00412m)^2
Now, we can calculate the area:
Area = 3.1416 × 0.00412^2
Area = 0.0532 m^2
Since the blood flows at a velocity of 35.0 cm/s, we need to convert this to meters:
Velocity = 35.0 cm/s = 0.35 m/s
Now, we can calculate the volume flow rate:
Volume flow rate = 0.0532 m^2 × 0.35 m/s
So, the volume flow rate in the aorta is 0.01862 m^3/s.
Now, let's calculate the total cross-sectional area of the capillaries. We can use the equation:
Volume flow rate = Area × Velocity
We know the volume flow rate is the same as in the aorta since all the blood flows from the aorta into the capillaries. The velocity through the capillaries is given as 1.10 mm/s, which we need to convert to meters:
Velocity = 1.10 mm/s = 0.0011 m/s
Rearranging the equation, we have:
Area = Volume flow rate / Velocity
Plugging in the values, we get:
Area = 0.01862 m^3/s / 0.0011 m/s
So, the total cross-sectional area of the capillaries is approximately 16.93 m^2.