Question

circle a has a radius of 4.5 in and \\(m\angle cab = 3\\) radians, what is the length of \\(\widehat{bc}\\)?

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Arc Length Formula"
],
"current_concepts": [
"Arc Length Formula",
"Radius of a Circle"
]
}
</pre_analysis>

<reasoning>

Identify the given values

\[

$$\begin{aligned} r &= 4.5\text{ in} \\ \theta &= 3\text{ radians} \end{aligned}$$

\]

Apply the arc length formula

\[

$$\begin{aligned} s &= r \cdot \theta \\ s &= 4.5 \cdot 3 \\ s &= 13.5\text{ in} \end{aligned}$$

\]
</reasoning>

<answer>
\(13.5\text{ in}\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Arc Length Formula"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Arc Length Formula"
],
"current_concepts": [
"Arc Length Formula",
"Radius of a Circle"
]
}
</pre_analysis>

<reasoning>

Identify the given values

\[

$$\begin{aligned} r &= 4.5\text{ in} \\ \theta &= 3\text{ radians} \end{aligned}$$

\]

Apply the arc length formula

\[

$$\begin{aligned} s &= r \cdot \theta \\ s &= 4.5 \cdot 3 \\ s &= 13.5\text{ in} \end{aligned}$$

\]
</reasoning>

<answer>
\(13.5\text{ in}\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Arc Length Formula"
]
}
</post_analysis>