According to Poiseuille's law, the resistance to flow of a...

According to Poiseuille's law, the resistance to flow of a blood vessel, $R,$ is directly proportional to the length, $l,$ and inversely proportional to the fourth power of the radius, $r .$ If $R=25$ when $l=12$ and $r=.2,$ find $R$ to the nearest hundredth as $r$ increases to $.3,$ while $l$ is unchanged.

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Key Concepts

Poiseuille's Law

Poiseuille's law is a principle in fluid dynamics that describes how the resistance to fluid flow within a cylindrical vessel (such as a blood vessel) is affected by its physical dimensions. It states that the resistance is directly proportional to the vessel's length and inversely proportional to the fourth power of its radius, reflecting how even small changes in radius can dramatically affect flow resistance.

Direct and Inverse Proportionality

Direct proportionality means that one variable increases as another increases, while inverse proportionality means that one variable decreases as the other increases. In this context, the vessel's resistance increases with its length (direct proportionality) and decreases very rapidly as the radius increases (due to the inverse fourth power relationship), highlighting the sensitivity of resistance to changes in vessel radius.

Constant of Proportionality

In proportional relationships, a constant of proportionality is used to quantify the exact relationship between variables. In the application of Poiseuille's law, determining this constant using given conditions is crucial for calculating how resistance changes when other properties, like the radius, are altered.

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