Relaclena, en cada caso, la longitud de are con la información dada. $$ \square $$ $$ r=3 \mathrm{~cm} v \theta=\frac{n}{3} $$ $$ \square $$ 7.054 cm $$ r=5 \mathrm{~cm} \times \theta=\frac{3 \pi}{2} $$ $$ 12,043 \mathrm{~cm} $$ $$ r=2 \mathrm{~cm} y \theta=\frac{5 \pi}{4} $$ 15.499 cm $$ r=4 \mathrm{~cm} \vee \theta=222^{\circ} $$ $$ r=\frac{5}{2} \mathrm{~cm} v \theta=315^{\circ} $$ $$ 13,744 \mathrm{~cm} $$ $$ r=4,6 \mathrm{~cm} \text { y } \theta=150^{\circ} $$ $$ 23,562 \mathrm{~cm} $$

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Alright, I need to solve these problems involving the calculation of arc length given the radius and the angle. Let's go through each one step by step.

**Problem 1:**
Given:
- $$ r = 3 \, \text{cm} $$
- $$ \theta = \frac{n}{3} $$

First, I need to understand what $$ \theta = \frac{n}{3} $$ represents. It seems like $$ n $$ is a variable, but without additional information, it's unclear what $$ n $$ is. Maybe $$ n $$ is a specific value or a parameter that needs to be defined. For now, I'll assume $$ n $$ is a constant angle in radians. If $$ n $$ is in degrees, I'll need to convert it to radians to use the arc length formula.

The formula for arc length $$ s $$ is:
$$
s = r \times \theta
$$
If $$ \theta $$ is in radians, then:
$$
s = 3 \, \text{cm} \times \frac{n}{3} = n \, \text{cm}
$$
But since $$ n $$ is undefined, I can't compute a numerical value for the arc length. Maybe there's a missing piece of information here.

**Problem 2:**
Given:
- $$ r = 5 \, \text{cm} $$
- $$ \theta = \frac{3\pi}{2} $$

Here, $$ \theta $$ is given in radians, which is good. Plugging into the arc length formula:
$$
s = 5 \, \text{cm} \times \frac{3\pi}{2} = \frac{15\pi}{2} \, \text{cm} \approx 23.562 \, \text{cm}
$$
This matches the provided answer of $$ 12,043 \, \text{cm} $$, which seems unusually large. Maybe there's a typo in the problem statement or the answer.

**Problem 3:**
Given:
- $$ r = 2 \, \text{cm} $$
- $$ \theta = \frac{5\pi}{4} $$

Again, $$ \theta $$ is in radians. Calculating the arc length:
$$
s = 2 \, \text{cm} \times \frac{5\pi}{4} = \frac{10\pi}{4} = \frac{5\pi}{2} \, \text{cm} \approx 7.854 \, \text{cm}
$$
The provided answer is $$ 15.499 \, \text{cm} $$, which is double the calculated value. This discrepancy suggests that either the angle was doubled or there's an error in the problem statement.

**Problem 4:**
Given:
- $$ r = 4 \, \text{cm} $$
- $$ \theta = 222^\circ $$

First, convert $$ \theta $$ to radians:
$$
\theta = 222^\circ \times \frac{\pi}{180} = \frac{222\pi}{180} = \frac{37\pi}{30} \, \text{radians}
$$
Now, calculate the arc length:
$$
s = 4 \, \text{cm} \times \frac{37\pi}{30} = \frac{148\pi}{30} \approx 15.562 \, \text{cm}
$$
The provided answer is $$ 13,744 \, \text{cm} $$, which is significantly larger. This indicates a possible error in the problem or the answer.

**Problem 5:**
Given:
- $$ r = \frac{5}{2} \, \text{cm} $$
- $$ \theta = 315^\circ $$

Convert $$ \theta $$ to radians:
$$
\theta = 315^\circ \times \frac{\pi}{180} = \frac{315\pi}{180} = \frac{7\pi}{4} \, \text{radians}
$$
Calculate the arc length:
$$
s = \frac{5}{2} \, \text{cm} \times \frac{7\pi}{4} = \frac{35\pi}{8} \approx 13.744 \, \text{cm}
$$
This matches the provided answer of $$ 13,744 \, \text{cm} $$, which seems unusually large. It's possible that the angle was intended to be in radians, leading to this discrepancy.

**Problem 6:**
Given:
- $$ r = 4.6 \, \text{cm} $$
- $$ \theta = 150^\circ $$

Convert $$ \theta $$ to radians:
$$
\theta = 150^\circ \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6} \, \text{radians}
$$
Calculate the arc length:
$$
s = 4.6 \, \text{cm} \times \frac{5\pi}{6} = \frac{23\pi}{6} \approx 12.043 \, \text{cm}
$$
The provided answer is $$ 23,562 \, \text{cm} $$, which is much larger. This suggests a possible error in the problem or the answer.

**Summary:**
- **Problem 1:** Cannot determine the arc length without additional information.
- **Problem 2:** Arc length is approximately $$ 23.562 \, \text{cm} $$, but the provided answer is $$ 12,043 \, \text{cm} $$.
- **Problem 3:** Arc length is approximately $$ 7.854 \, \text{cm} $$, but the provided answer is $$ 15.499 \, \text{cm} $$.
- **Problem 4:** Arc length is approximately $$ 15.562 \, \text{cm} $$, but the provided answer is $$ 13,744 \, \text{cm} $$.
- **Problem 5:** Arc length is approximately $$ 13.744 \, \text{cm} $$, but the provided answer is $$ 13,744 \, \text{cm} $$.
- **Problem 6:** Arc length is approximately $$ 12.043 \, \text{cm} $$, but the provided answer is $$ 23,562 \, \text{cm} $$.

It appears there are inconsistencies or possible errors in the provided answers for Problems 2, 3, 4, 5, and 6. It's essential to verify the given information or check for any typos in the problem statements.

No se puede determinar la longitud del arco en el primer problema sin información adicional. En los problemas 2, 3, 4, 5 y 6, los resultados calculados no coinciden con los proporcionados, lo que sugiere posibles errores en los datos o en las respuestas proporcionadas.

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