In intravenous feeding, a needle is inserted in a vein in the patient's arm and a tube leads from the needle to a reservoir of fluid (density 1050 kg/m³) located at height h above the arm. The top of the reservoir is open to the air. If the gauge pressure inside the vein is 5980 Pa, what is the minimum value of h that allows fluid to enter the vein? Assume the needle diameter is large enough that you can ignore the viscosity of the liquid.

검증된 단계별 안내

Understand that the fluid will flow into the vein if the pressure at the needle is greater than the gauge pressure inside the vein. The gauge pressure is the pressure relative to atmospheric pressure.

The pressure at the needle is due to the height of the fluid column above the needle. This pressure can be calculated using the hydrostatic pressure formula: \( P = \rho g h \), where \( \rho \) is the fluid density, \( g \) is the acceleration due to gravity, and \( h \) is the height of the fluid column.

Set up the equation to find the minimum height \( h \) by equating the hydrostatic pressure to the gauge pressure inside the vein: \( \rho g h = 5980 \) Pa.

Substitute the given values into the equation: \( \rho = 1050 \) kg/m³ and \( g = 9.81 \) m/s². This gives \( 1050 \times 9.81 \times h = 5980 \).

Solve for \( h \) by rearranging the equation: \( h = \frac{5980}{1050 \times 9.81} \). This will give you the minimum height required for the fluid to enter the vein.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Fluid Pressure

Fluid pressure is the force exerted by a fluid per unit area. In this context, the pressure exerted by the fluid in the reservoir must overcome the gauge pressure in the vein for fluid to flow into it. The pressure due to the fluid column is calculated using the formula P = ρgh, where ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column.

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It is the pressure inside the vein that the fluid from the reservoir must overcome to enter the vein. In this problem, the gauge pressure is given as 5980 Pa, which means the fluid pressure from the reservoir must be at least this value for fluid to flow into the vein.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in the fluid. In this scenario, the hydrostatic pressure from the fluid column in the reservoir is what drives the fluid into the vein. The height h of the fluid column determines the hydrostatic pressure, which must be sufficient to overcome the vein's gauge pressure.