Question

find the length of arc abc.
use 3.14 for the value of \\(\pi\\).

? cm
round to the nearest hundredth of a cm.

Explanation:

<pre_analysis>
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"step_cot",
"blank_filling_formatter",
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"concepts_used": [],
"new_concepts": [
"Arc Length Formula",
"Central Angle Theorem"
],
"current_concepts": [
"Arc Length Formula",
"Central Angle Theorem"
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}
</pre_analysis>

<reasoning>

Identify the given values

\[

$$\begin{aligned} r &= 54\text{ cm} \\ \theta &= 240^\circ \\ \pi &\approx 3.14 \end{aligned}$$

\]

Apply the arc length formula

\[

$$\begin{aligned} s &= \frac{\theta}{360^\circ} \cdot 2\pi r \\ s &= \frac{240}{360} \cdot 2 \cdot 3.14 \cdot 54 \end{aligned}$$

\]

Calculate the final value

\[

$$\begin{aligned} s &= \frac{2}{3} \cdot 108 \cdot 3.14 \\ s &= 72 \cdot 3.14 \\ s &= 226.08\text{ cm} \end{aligned}$$

\]
</reasoning>

<answer>
Find the length of arc ABC. Use 3.14 for the value of \(\pi\).

<blank>226.08</blank> cm

Round to the nearest hundredth of a cm.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Arc Length Formula"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Arc Length Formula",
"Central Angle Theorem"
],
"current_concepts": [
"Arc Length Formula",
"Central Angle Theorem"
]
}
</pre_analysis>

<reasoning>

Identify the given values

\[

$$\begin{aligned} r &= 54\text{ cm} \\ \theta &= 240^\circ \\ \pi &\approx 3.14 \end{aligned}$$

\]

Apply the arc length formula

\[

$$\begin{aligned} s &= \frac{\theta}{360^\circ} \cdot 2\pi r \\ s &= \frac{240}{360} \cdot 2 \cdot 3.14 \cdot 54 \end{aligned}$$

\]

Calculate the final value

\[

$$\begin{aligned} s &= \frac{2}{3} \cdot 108 \cdot 3.14 \\ s &= 72 \cdot 3.14 \\ s &= 226.08\text{ cm} \end{aligned}$$

\]
</reasoning>

<answer>
Find the length of arc ABC. Use 3.14 for the value of \(\pi\).

<blank>226.08</blank> cm

Round to the nearest hundredth of a cm.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Arc Length Formula"
]
}
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