Question

The radius of the aorta is «10 mm and the blood flowing through it has a speed of about 300 mm/s. A capillary has a radius of about 4ˆ10´3 mm but there are literally billions of them. The average speed of blood through the capillaries is about 5ˆ10´4 m/s. (i) Calculate the effective cross sectional area of the capillaries and (ii) the approximate number of capillaries.

Answer 1

Answer:

(I). The effective cross sectional area of the capillaries is 0.188 m².

(II). The approximate number of capillaries is $3.74\times10^{9}$

Explanation:

Given that,

Radius of aorta = 10 mm

Speed = 300 mm/s

Radius of capillary $r=4\times10^{-3}\ mm$

Speed of blood $v=5\times10^{-4}\ m/s$

(I). We need to calculate the effective cross sectional area of the capillaries

Using continuity equation

$A_{1}v_{1}=A_{2}v_{2}$

Where. v₁ = speed of blood in capillarity

A₂ = area of cross section of aorta

v₂ =speed of blood in aorta

Put the value into the formula

$A_{1}=A_{2}\times\dfrac{v_{2}}{v_{1}}$

$A_{1}=\pi\times(10\times10^{-3})^2\times\dfrac{300\times10^{-3}}{5\times10^{-4}}$

$A_{1}=0.188\ m^2$

(II). We need to calculate the approximate number of capillaries

Using formula of area of cross section

$A_{1}=N\pi r_{c}^2$

$N=\dfrac{A_{1}}{\pi\times r_{c}^2}$

Put the value into the formula

$N=\dfrac{0.188}{\pi\times(4\times10^{-6})^2}$

$N=3.74\times10^{9}$

Hence, (I). The effective cross sectional area of the capillaries is 0.188 m².

(II). The approximate number of capillaries is $3.74\times10^{9}$