With the distance: 0.610 m, nutrient solution (ρ = 1040 kg/m3) can just barely enter the blood in the vein. What is the gauge pressure of the venous blood? Express your answer in millimeters of mercury.

ñ * g * 0.610 m is the gaughe pressure of the blood entering the vein, in units of N/m^2.

To convert to mm of Hg, use the conversion 1 atm = 1.013*10^5 N/m^2 = 760 mm Hg
(or divide 610 mm by the ratio of the densities of mercury and the fluid, which is about 13.5)

To find the gauge pressure of the venous blood, we can use the concept of hydrostatic pressure. The hydrostatic pressure is given by the equation:

P = ρgh

Where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height or distance. In this case, the height or distance is given as 0.610 m and the density of the nutrient solution is 1040 kg/m^3.

First, let's convert the density from kg/m^3 to g/cm^3 to be consistent with the unit of millimeters of mercury (mmHg).

Density in g/cm^3 = 1040 kg/m^3 * (1 g/1000 kg) * (1 m/100 cm)^3
Density in g/cm^3 = 1.04 g/cm^3

Next, we need to convert the height or distance from meters to centimeters to match the density unit.

Height in cm = 0.610 m * 100 cm/m
Height in cm = 61 cm

Now, we can plug the values into the equation for hydrostatic pressure:

P = ρgh
P = (1.04 g/cm^3) * (9.8 m/s^2) * (61 cm)
P = 6003.24 g · cm/s^2

To convert the pressure units from g · cm/s^2 to mmHg, we need to use the following conversion factors:
1 g · cm/s^2 = 0.00001422 psi
1 psi = 51.715 mmHg

P = (6003.24 g · cm/s^2) * (0.00001422 psi / (g · cm/s^2)) * (51.715 mmHg / psi)
P ≈ 4.28 mmHg

Therefore, the gauge pressure of the venous blood is approximately 4.28 mmHg.