Question

find the area of the shaded sector. round to the nearest hundredth.

\\(145^\circ\\)
\\(18\text{ ft}\\)
\\(\text{ft}^2\\)

Explanation:

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"concepts_used": [],
"new_concepts": [
"Area of Sector"
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"Area of Sector"
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<reasoning>

Identify the given parameters

\[

$$\begin{aligned} r &= 18\text{ ft} \\ \theta &= 145^\circ \end{aligned}$$

\]

Apply the sector area formula

\[

$$\begin{aligned} A &= \frac{\theta}{360^\circ} \cdot \pi r^2 \\ A &= \frac{145}{360} \cdot \pi \cdot 18^2 \\ A &= \frac{29}{72} \cdot 324\pi \\ A &= 130.5\pi \end{aligned}$$

\]

Calculate the decimal value

\[

$$\begin{aligned} A &\approx 130.5 \cdot 3.14159265 \\ A &\approx 409.9778 \\ A &\approx 409.98 \end{aligned}$$

\]
</reasoning>

<answer>
Find the area of the shaded sector. Round to the nearest hundredth.

<blank>409.98</blank> \(ft^2\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Sector"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
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"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
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"concepts_used": [],
"new_concepts": [
"Area of Sector"
],
"current_concepts": [
"Area of Sector"
]
}
</pre_analysis>

<reasoning>

Identify the given parameters

\[

$$\begin{aligned} r &= 18\text{ ft} \\ \theta &= 145^\circ \end{aligned}$$

\]

Apply the sector area formula

\[

$$\begin{aligned} A &= \frac{\theta}{360^\circ} \cdot \pi r^2 \\ A &= \frac{145}{360} \cdot \pi \cdot 18^2 \\ A &= \frac{29}{72} \cdot 324\pi \\ A &= 130.5\pi \end{aligned}$$

\]

Calculate the decimal value

\[

$$\begin{aligned} A &\approx 130.5 \cdot 3.14159265 \\ A &\approx 409.9778 \\ A &\approx 409.98 \end{aligned}$$

\]
</reasoning>

<answer>
Find the area of the shaded sector. Round to the nearest hundredth.

<blank>409.98</blank> \(ft^2\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Sector"
]
}
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