The flow rate of a fluid is expressed as a velocity of fluid multiplied by the cross-sectional area of the pipe (vA). Assume that the pipe is a blood vessel with circular cross-sectional area calculated as A = ðr², where r is radius. In a continuous flow rate, where v₁A₁ = v₂A₂, what is the change of the velocity of blood if a blood vessel with radius r₁ reduced its radius by a factor of 2 to r₂ = r₁/2? A)v₂ = (1/4)v₁ B)v₂ = 2v₁ C)v₂ = 4v₁ D)v₂ = (1/2)v₁

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Quizplus · Accepted Answer

The area of a circle is proportional to the square of its radius ($A = \pi r^2$). If the radius is reduced by a factor of 2 ($r_2 = \frac{1}{2}r_1$), the new area ($A_2$) becomes $\frac{1}{4}$ of the original area ($A_1$), because $(\frac{1}{2})^2 = \frac{1}{4}$. To maintain a continuous flow rate ($v_1A_1 = v_2A_2$), if the area is reduced to $\frac{1}{4}$, the velocity must increase by a factor of 4 to compensate ($v_2 = 4v_1$).

The Flow Rate of a Fluid Is Expressed as a Velocity