Question

determine (a) the area and (b) the circumference of the circle.

a. the area of the circle is 572.40 \\(\text{ft}^2\\).
(type an integer or decimal rounded to the nearest hundredth as needed.)

Explanation:

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"concepts_used": [
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],
"new_concepts": [
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"current_concepts": [
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"Circumference of Circle"
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</pre_analysis>

<reasoning>

Find the radius of the circle

Using the Area of Circle knowledge point
\[

$$\begin{aligned} &d = 27\text{ ft}\\ &r = \frac{d}{2} = \frac{27}{2} = 13.5\text{ ft} \end{aligned}$$

\]

Calculate the area of the circle

Using the Area of Circle knowledge point
\[

$$\begin{aligned} &A = \pi r^2 = \pi (13.5)^2 = 182.25\pi\text{ ft}^2\\ &A \approx 182.25 \times 3.14159265 \approx 572.56\text{ ft}^2 \end{aligned}$$

\]

Calculate the circumference of the circle

We use the circumference formula:
\[
C = \pi d
\]
Substitute the given diameter:
\[
C = 27\pi\text{ ft}
\]
Calculate the decimal value:
\[
C \approx 27 \times 3.14159265 \approx 84.82\text{ ft}
\]
</reasoning>

<answer>

Question 1

The area of the circle is <blank>\(572.56\)</blank> \(\text{ft}^2\).

Question 2

The circumference of the circle is <blank>\(84.82\)</blank> \(\text{ft}\).
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circumference of Circle"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
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"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
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"concepts_used": [
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],
"new_concepts": [
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],
"current_concepts": [
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"Circumference of Circle"
]
}
</pre_analysis>

<reasoning>

Find the radius of the circle

Using the Area of Circle knowledge point
\[

$$\begin{aligned} &d = 27\text{ ft}\\ &r = \frac{d}{2} = \frac{27}{2} = 13.5\text{ ft} \end{aligned}$$

\]

Calculate the area of the circle

Using the Area of Circle knowledge point
\[

$$\begin{aligned} &A = \pi r^2 = \pi (13.5)^2 = 182.25\pi\text{ ft}^2\\ &A \approx 182.25 \times 3.14159265 \approx 572.56\text{ ft}^2 \end{aligned}$$

\]

Calculate the circumference of the circle

We use the circumference formula:
\[
C = \pi d
\]
Substitute the given diameter:
\[
C = 27\pi\text{ ft}
\]
Calculate the decimal value:
\[
C \approx 27 \times 3.14159265 \approx 84.82\text{ ft}
\]
</reasoning>

<answer>

Question 1

The area of the circle is <blank>\(572.56\)</blank> \(\text{ft}^2\).

Question 2

The circumference of the circle is <blank>\(84.82\)</blank> \(\text{ft}\).
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circumference of Circle"
]
}
</post_analysis>