during blood transfusion the needle is inserted into a vein where guage pressure is 2000 pa.at what height must the blood container be placed so that blood may jst entr d vein.d=1 * 10 to power of minus 3.

To determine the height at which the blood container must be placed for the blood to enter the vein during a transfusion, we can use the principles of fluid mechanics. Specifically, we will apply the hydrostatic pressure formula, which relates the pressure difference to the height of a fluid column. Let's break this down step by step.

Understanding the Pressure Relationship

The pressure exerted by a fluid column can be calculated using the formula:

P = ρgh

Where:

• P is the pressure (in pascals, Pa)
• ρ is the density of the fluid (in kg/m³)
• g is the acceleration due to gravity (approximately 9.81 m/s²)
• h is the height of the fluid column (in meters)

Given Values

From your question, we know:

• The gauge pressure in the vein is 2000 Pa.
• The density of blood is approximately 1 x 10^-3 kg/m³ (which seems to be a misunderstanding; the density of blood is typically around 1060 kg/m³, but we'll proceed with your value for calculation purposes).
• The acceleration due to gravity, g, is 9.81 m/s².

Calculating the Height

We need to rearrange the hydrostatic pressure formula to solve for height h:

h = P / (ρg)

Now, substituting the values we have:

h = 2000 Pa / (1 x 10^-3 kg/m³ * 9.81 m/s²)

Performing the Calculation

Calculating the denominator:

1 x 10^-3 kg/m³ * 9.81 m/s² = 0.00981 kg/(m·s²) = 0.00981 N/m²

Now substituting back into the height equation:

h = 2000 Pa / 0.00981 N/m²

h ≈ 203,000.2 m

Interpreting the Result

This height is impractically large, indicating that the density value used was likely incorrect. The actual density of blood is around 1060 kg/m³. Let’s recalculate using this correct value:

h = 2000 Pa / (1060 kg/m³ * 9.81 m/s²)

Calculating the denominator again:

1060 kg/m³ * 9.81 m/s² ≈ 10,396.6 N/m²

Now substituting back into the height equation:

h = 2000 Pa / 10,396.6 N/m²

h ≈ 0.192 m

Final Thoughts

Thus, the blood container should be placed approximately 0.192 meters (or about 19.2 centimeters) above the level of the vein for the blood to flow into it effectively. This calculation highlights the importance of using accurate values for density in fluid mechanics problems.