The aorta carries blood away from the heart at a speed of about 43 cm/s and has a radius of approximately 1.1 cm. The aorta branches eventually into a large number of tiny capillaries that distribute the blood to the various body organs. In a capillary, the blood speed is approximately 0.066 cm/s, and the radius is about 5.5 x 10-4 cm. Treat the blood as an incompressible fluid, and use these data to determine the approximate number of capillaries in the human body.
Aa/Ac - area of the aorta/area of capillary
Va/Vc - velocity of blood from the aorta/velocity of blood from capillaries
ac - area of a single capillary
ra/rc - radius of aorta/radius of capillary
n - number of capillaries
AaVa = AcVc
AaVa = nacVc. Separate area
pi*ra^2*Va = n*pi*rc^2*Vc. Arrange to solve for n
N = (ra^2*Va)/(rc^2*Vc) = 2.6 * 10^9 capillaries
Q = flow rate = speed * area
43 pi R^2 = n * .066 pi r^2
To determine the approximate number of capillaries in the human body, we can make use of the principle of conservation of mass. Since blood is an incompressible fluid, the volume flow rate at any point in the circulatory system must be constant.
The volume flow rate (Q) is given by the equation:
Q = A * v
where Q is the volume flow rate, A is the cross-sectional area of the vessel, and v is the speed of the blood.
Let's start by calculating the cross-sectional area of the aorta:
Aorta cross-sectional area (A_aorta) = π * (radius_aorta)^2
= π * (1.1 cm)^2
Now, we can calculate the volume flow rate in the aorta:
Q_aorta = A_aorta * v_aorta
= π * (1.1 cm)^2 * 43 cm/s
Next, let's calculate the volume flow rate in the capillary:
Q_capillary = A_capillary * v_capillary
= π * (5.5 x 10^(-4) cm)^2 * 0.066 cm/s
Since the volume flow rate is constant, Q_aorta = Q_capillary. This implies:
A_aorta * v_aorta = A_capillary * v_capillary
We can rearrange the equation to solve for the ratio of the areas:
A_capillary / A_aorta = v_aorta / v_capillary
Now, substitute the given values into the equation:
A_capillary / (π * (1.1 cm)^2) = (43 cm/s) / (0.066 cm/s)
Solving for A_capillary, we have:
A_capillary = (π * (1.1 cm)^2) * (0.066 cm/s) / (43 cm/s)
Finally, we can use the area of a single capillary to calculate the approximate number of capillaries in the human body.
Assuming the average human body contains about 5 liters of blood, we can calculate the total volume of capillaries:
Total volume of capillaries = volume of blood = 5 L = 5000 cm^3
Assuming the capillaries have a uniform radius, we can calculate the total surface area of the capillaries:
Total surface area = A_capillary * number of capillaries
Therefore,
number of capillaries = Total surface area / A_capillary
Substitute the values into the equation:
number of capillaries = (Total volume of capillaries) / (A_capillary)
After performing the calculations, you should arrive at an approximate number of capillaries in the human body.