Let's go through each part of the problem provided:

Given:

Poiseuille's Law for the rate of fluid flow is expressed as: $F = cR^4$ where:

• $F$ is the flow rate (volume per unit time),
• $c$ is a constant,
• $R$ is the radius of the tube.

Solution to each part:

(a) Increase in Flow Rate $F$ when Radius $R$ Increases by 10%

If $R$ increases by 10%, then the new radius $R_{\text{new}}$ is: $R_{\text{new}} = 1.10 \times R$

Since $F = cR^4$, the new flow rate $F_{\text{new}}$ is: $F_{\text{new}} = c(R_{\text{new}})^4 = c(1.10 \cdot R)^4$

Calculating $(1.10)^4$: $(1.10)^4 \approx 1.4641$

Thus: $F_{\text{new}} = 1.4641 \cdot cR^4 = 1.4641 \cdot F$

This shows that $F$ increases by approximately 46.41% when $R$ increases by 10%.

(b) Comparing Flow Rates for Different Pipe Radii (3/4-inch vs. 1/2-inch)

Let:

• Radius $R_1 = \frac{3}{4}$ inch
• Radius $R_2 = \frac{1}{2}$ inch

The flow rate $F_1$ for the 3/4-inch pipe: $F_1 = c \left(\frac{3}{4}\right)^4 = c \cdot \frac{81}{256}$

The flow rate $F_2$ for the 1/2-inch pipe: $F_2 = c \left(\frac{1}{2}\right)^4 = c \cdot \frac{1}{16}$

To find the ratio $\frac{F_1}{F_2}$: $\frac{F_1}{F_2} = \frac{\frac{81}{256}}{\frac{1}{16}} = \frac{81 \times 16}{256} = \frac{1296}{256} = 5.0625$

So, the flow rate through the 3/4-inch pipe is approximately 5.06 times greater than that through the 1/2-inch pipe.

(c) Percentage of Blood Flow through a Partially Blocked Artery (Radius Halved)

If an artery’s radius is halved, the new radius $R_{\text{blocked}}$ is: $R_{\text{blocked}} = \frac{1}{2} R$

The flow rate $F_{\text{blocked}}$ through the blocked artery is: $F_{\text{blocked}} = c \left(\frac{1}{2} R\right)^4 = c \cdot \frac{1}{16} R^4 = \frac{1}{16} F$

Thus, only 6.25% of the usual blood flow will flow through the partially blocked artery.

Would you like further details on any part of these calculations or have any questions?

Related Questions:

1. How does Poiseuille’s law change if the radius of the tube triples?
2. What is the flow rate ratio if the radius of the pipe changes from 1 inch to 2 inches?
3. How sensitive is the flow rate $F$ to small changes in radius $R$?
4. Can Poiseuille’s law apply to non-circular cross-sectional tubes?
5. What impact does fluid viscosity have in Poiseuille’s Law?

Tip:

Remember, in Poiseuille’s Law, the flow rate $F$ is highly sensitive to the radius $R$ due to the fourth power, so small changes in $R$ can greatly affect $F$.