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The problem in the image is related to fluid dynamics in blood vessels. Here's a breakdown of the problem:

- A blood vessel with radius \( r \) splits into four smaller vessels, each with radius \( \frac{r}{3} \).
- The average velocity in the widest vessel is \( v \).
- The question asks: What is the average velocity in each of the narrower vessels?

To solve this, we can use the principle of conservation of flow rate (continuity equation). The flow rate \( Q \) is given by:

\[
Q = A \cdot v
\]

Where:
- \( A \) is the cross-sectional area of the vessel, which is proportional to \( r^2 \) (since \( A = \pi r^2 \) for a circular cross-section),
- \( v \) is the velocity of the fluid.

The total flow rate through the large vessel must equal the total flow rate through the smaller vessels:

\[
Q_{\text{large}} = Q_{\text{small}}
\]

For the large vessel:

\[
Q_{\text{large}} = A_{\text{large}} \cdot v = \pi r^2 \cdot v
\]

For the four smaller vessels, each has a radius of \( \frac{r}{3} \), so the area of each is:

\[
A_{\text{small}} = \pi \left( \frac{r}{3} \right)^2 = \frac{\pi r^2}{9}
\]

The total flow rate through the four small vessels is:

\[
Q_{\text{small}} = 4 \cdot A_{\text{small}} \cdot v_{\text{small}} = 4 \cdot \frac{\pi r^2}{9} \cdot v_{\text{small}} = \frac{4 \pi r^2}{9} \cdot v_{\text{small}}
\]

Equating the two flow rates:

\[
\pi r^2 \cdot v = \frac{4 \pi r^2}{9} \cdot v_{\text{small}}
\]

Canceling out \( \pi r^2 \) on both sides:

\[
v = \frac{4}{9} \cdot v_{\text{small}}
\]

Solving for \( v_{\text{small}} \):

\[
v_{\text{small}} = \frac{9}{4} \cdot v
\]

Thus, the average velocity in each of the smaller vessels is \( \frac{9}{4} v \).

---

Would you like further clarification or details on any step of the solution? Here are some related questions:

1. How does the principle of conservation of mass apply to fluid dynamics?
2. What happens to the velocity if the radius of the vessel increases instead of decreases?
3. How is the area of a vessel related to its radius?
4. What assumptions are made in this problem regarding fluid flow?
5. What is the significance of the continuity equation in blood flow through vessels?

**Tip**: In fluid dynamics, decreasing the radius of a vessel significantly increases the velocity of the fluid if the flow rate remains constant.

Un vaso sanguíneo de radio r se divide en cuatro vasos sanguíneos, cada uno de radio r/3. Si la rapidez media en el vaso más ancho es v, ¿cuál es la rapidez media en cada uno de los vasos estrechos?

Learn how to calculate the average velocity in smaller blood vessels when a large vessel with radius r splits into four smaller vessels with radius r/3. Using the continuity equation and the principle of conservation of flow rate, we solve for the velocity in the narrower vessels.

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