Question

Estimate the average blood pressure in a person's foot, if the foot is \(1.37 \mathrm{m}\) below the aorta, where the average blood pressure is 104 mm Hg. For the purposes of this estimate, assume the blood isn't flowing.

Short answer

The average blood pressure in the foot is approximately 210.7 mm Hg.

Step-by-step solution

Understanding Hydrostatic Pressure

The hydrostatic pressure difference due to a height difference can be calculated using the formula: \[ \Delta P = \rho g h \]where \(\rho\) is the density of blood (approximately 1060 kg/m³), \(g\) is the acceleration due to gravity (9.81 m/s²), and \(h\) is the height difference (1.37 m).

Calculate the Hydrostatic Pressure Difference

Substitute the given values into the equation: \[ \Delta P = 1060 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 1.37 \, \text{m} \]Calculate \(\Delta P\):\[ \Delta P \approx 14231 \, \text{Pa} \, (or \, N/m}^2) \].

Convert Pressure Difference to mm Hg

To convert the pressure difference from pascals to mm Hg, use the conversion factor \(1 \, \text{mm Hg} \approx 133.322 \, \text{Pa}\):\[ \Delta P = \frac{14231 \, \text{Pa}}{133.322 \, \text{Pa/mm Hg}} \approx 106.7 \, \text{mm Hg} \].

Add the Hydrostatic Pressure to Initial Pressure

Since the blood in the foot is below the aorta, add this hydrostatic pressure to the initial aortic pressure of 104 mm Hg:\[ P_{\text{foot}} = 104 \, \text{mm Hg} + 106.7 \, \text{mm Hg} = 210.7 \, \text{mm Hg} \].

Key concepts

Blood Pressure Calculation

Blood pressure calculation is an essential aspect of understanding how pressure varies at different points in the body, particularly when the position changes, like from the aorta to the foot. To estimate the blood pressure at a point lower than the heart, like a foot, you need to consider both the initial pressure and the hydrostatic pressure due to the weight of the blood column above it.

In our example, the foot is 1.37 meters below the aorta. The standard atmospheric pressure is 104 mm Hg at the aorta. To find the blood pressure in the foot, we calculate the hydrostatic pressure difference. This difference occurs because the blood exerts pressure due to gravity over the vertical distance it travels. Hydrostatic pressure is calculated using the formula \( \Delta P = \rho g h \). Substituting in the given values, we find that this pressure difference is approximately 14231 Pa, which increases the effective pressure exerted at the foot.

Finally, the pressure in the foot is obtained by adding this hydrostatic pressure difference to the initial aortic pressure, resulting in a total foot pressure of about 210.7 mm Hg. This calculation is crucial for understanding how the body manages blood flow and pressure to ensure every part receives adequate oxygen and nutrients.

Pressure Conversion

Pressure conversion is a fundamental process in physics and engineering because pressure is often expressed in various units. Converting between these units helps in solving problems and understanding results in a standard format.

In many scientific contexts, pressure is measured in pascals (Pa), but when it comes to medicine, such as measuring blood pressure, it's commonly measured in millimeters of mercury (mm Hg). Each unit has its significance: pascal as the SI unit, and mm Hg for ease of measurement and tradition in medical practices.

• 1 Pascal is very small compared to mm Hg.
• 133.322 Pascal equals 1 mm Hg.

When converting, you adjust the values through multiplication or division using these conversion factors. For example, a hydrostatic pressure of 14231 Pa converts to approximately 106.7 mm Hg. Performing these conversions accurately ensures that measurements remain consistent across different systems and calculations.

Pascal to mm Hg Conversion

The conversion between pascals and millimeters of mercury is crucial in many practical scenarios, especially in medicine and physics. This conversion allows pressures expressed in pascals to be accurately translated into mm Hg, supporting their use in contexts like medical exams where blood pressure is monitored.

To convert from pascals to mm Hg, you divide the pressure value in pascals by the conversion factor of 133.322 Pa/mm Hg. This conversion factor is derived from the relationship between the weight of a column of mercury and the pressure it exerts at its base.

• Pascal (Pa) is the SI unit of pressure: 1 Pa = 1 N/m².
• Millimeters of mercury (mm Hg) is a non-SI unit accepted for use with the SI: 1 mm Hg is approximately 133.322 Pa.
• This conversion factor allows seamless transition between the units, facilitating compatibility in various fields.

Correctly converting pressure helps maintain accuracy in medical diagnostics and treatment, ensuring that blood pressure readings are accurate and comparable despite the units used during measurement.