Question

find the area of the larger sector.
round to the nearest tenth.

area = \\( ? \text{ miles}^2 \\)

Explanation:

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"concepts_used": [
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<reasoning>

Identify the given values

\[

$$\begin{aligned} r &= 13.4\text{ miles} \\ \theta &= 255^\circ \end{aligned}$$

\]

Apply the sector area formula

\[

$$\begin{aligned} \text{Area} &= \frac{\theta}{360^\circ} \cdot \pi r^2 \\ \text{Area} &= \frac{255}{360} \cdot \pi \cdot (13.4)^2 \end{aligned}$$

\]

Calculate the numerical value

\[

$$\begin{aligned} \text{Area} &= \frac{17}{24} \cdot \pi \cdot 179.56 \\ \text{Area} &\approx 0.708333 \cdot 3.14159265 \cdot 179.56 \\ \text{Area} &\approx 399.598 \approx 399.6 \end{aligned}$$

\]
</reasoning>

<answer>
Find the area of the larger sector. Round to the nearest tenth.

Area = <blank>399.6</blank> \(\text{miles}^2\)
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Sector Area Formula"
]
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Answer:

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"concepts_used": [
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],
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</pre_analysis>

<reasoning>

Identify the given values

\[

$$\begin{aligned} r &= 13.4\text{ miles} \\ \theta &= 255^\circ \end{aligned}$$

\]

Apply the sector area formula

\[

$$\begin{aligned} \text{Area} &= \frac{\theta}{360^\circ} \cdot \pi r^2 \\ \text{Area} &= \frac{255}{360} \cdot \pi \cdot (13.4)^2 \end{aligned}$$

\]

Calculate the numerical value

\[

$$\begin{aligned} \text{Area} &= \frac{17}{24} \cdot \pi \cdot 179.56 \\ \text{Area} &\approx 0.708333 \cdot 3.14159265 \cdot 179.56 \\ \text{Area} &\approx 399.598 \approx 399.6 \end{aligned}$$

\]
</reasoning>

<answer>
Find the area of the larger sector. Round to the nearest tenth.

Area = <blank>399.6</blank> \(\text{miles}^2\)
</answer>

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