Question

a company produces individually wrapped spherical cookies. the minimum amount of wrapping material needed to cover a cookie, assuming no edges overlap, is 314 square inches. what is the diameter of one cookie? (use 3.14 for the value of \\(\pi\\).)

5 in.
10 in.
20 in.
25 in.

Explanation:

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</pre_analysis>

<reasoning>

Set up the surface area equation for a sphere

\[

$$\begin{aligned} A &= 4\pi r^2 \\ 314 &= 4(3.14)r^2 \end{aligned}$$

\]

Solve for the radius

\[

$$\begin{aligned} 314 &= 12.56r^2 \\ r^2 &= \frac{314}{12.56} \\ r^2 &= 25 \\ r &= 5 \end{aligned}$$

\]

Calculate the diameter

\[

$$\begin{aligned} d &= 2r \\ d &= 2(5) \\ d &= 10\text{ in.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>5 in.</mcq-option>
<mcq-correct>10 in.</mcq-correct>
<mcq-option>20 in.</mcq-option>
<mcq-option>25 in.</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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]
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Answer:

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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
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"concepts_used": [],
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"Surface Area of a Sphere",
"Sphere Diameter"
],
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"Sphere Diameter"
]
}
</pre_analysis>

<reasoning>

Set up the surface area equation for a sphere

\[

$$\begin{aligned} A &= 4\pi r^2 \\ 314 &= 4(3.14)r^2 \end{aligned}$$

\]

Solve for the radius

\[

$$\begin{aligned} 314 &= 12.56r^2 \\ r^2 &= \frac{314}{12.56} \\ r^2 &= 25 \\ r &= 5 \end{aligned}$$

\]

Calculate the diameter

\[

$$\begin{aligned} d &= 2r \\ d &= 2(5) \\ d &= 10\text{ in.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>5 in.</mcq-option>
<mcq-correct>10 in.</mcq-correct>
<mcq-option>20 in.</mcq-option>
<mcq-option>25 in.</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Surface Area of a Sphere"
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