Poiseuille's law states that the resistance R for blood...

Poiseuille's law states that the resistance $R$ for blood flowing in a blood vessel varies directly as the length $L$ of the vessel and inversely as the fourth power of its radius $r$. Stated as an equation, $$ R(L, r)=k \frac{L}{r^{4}} \quad k \text { a constant } $$ Find $R(8,1)$ and $R(4,0.2)$.

Poiseuille's law states that the resistance $R$ for blood flowing in a blood vessel varies directly as the length $L$ of the vessel and inversely as the fourth power of its radius $r$. Stated as an equation,
$$
R(L, r)=k \frac{L}{r^{4}} \quad k \text { a constant }
$$
Find $R(8,1)$ and $R(4,0.2)$.

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Key Concepts

Constant of Proportionality

The constant of proportionality, often denoted as k, is a fixed value that ensures the equation accurately represents the relationship between the quantities involved. It scales the relationship to reflect specific units or empirical measurements within the physical system.

Exponentiation

Exponentiation is the mathematical operation of raising a number to a power, which in this law highlights the dramatic impact of the vessel's radius on resistance. The fourth power dependence indicates that changes in radius are amplified, affecting the flow resistance non-linearly.

Direct Variation

Direct variation refers to a relationship between two quantities in which one quantity increases (or decreases) in exact proportion to another. In the context of fluid dynamics, an increase in the length of the blood vessel leads to a proportional increase in resistance.

Inverse Variation

Inverse variation describes a relation where an increase in one variable leads to a decrease in another, according to a specific power. Here, the resistance is inversely proportional to the fourth power of the vessel's radius, meaning that even small decreases in the radius cause a significant increase in resistance.

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