Question

the table below shows the number of days, \\(y\\), needed to complete a project as a function of the number of full-time staff, \\(x\\), working on the project. which rational function best models the data in the table?

\

$$\begin{array}{|c|c|} \\hline \\text{people working, } x & \\text{days, } y \\\\ \\hline 36 & 2 \\\\ \\hline 18 & 4 \\\\ \\hline 8 & 9 \\\\ \\hline 6 & 12 \\\\ \\hline \\end{array}$$

\\(y = \frac{x}{72}\\)

\\(y = \frac{x}{18}\\)

\\(y = \frac{18}{x}\\)

\\(y = \frac{72}{x}\\)

Explanation:

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"multiple_choice_formatter",
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],
"new_concepts": [],
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<reasoning>

Identify the relationship between variables

\[

$$\begin{aligned} &x \cdot y = 36 \cdot 2 = 72 \\ &x \cdot y = 18 \cdot 4 = 72 \\ &x \cdot y = 8 \cdot 9 = 72 \\ &x \cdot y = 6 \cdot 12 = 72 \end{aligned}$$

\]

Solve for y

\[
y = \frac{72}{x}
\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \frac{x}{72}\)</mcq-option>
<mcq-option>(B) \(y = \frac{x}{18}\)</mcq-option>
<mcq-option>(C) \(y = \frac{18}{x}\)</mcq-option>
<mcq-correct>(D) \(y = \frac{72}{x}\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Inverse Variation"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
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],
"new_concepts": [],
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</pre_analysis>

<reasoning>

Identify the relationship between variables

\[

$$\begin{aligned} &x \cdot y = 36 \cdot 2 = 72 \\ &x \cdot y = 18 \cdot 4 = 72 \\ &x \cdot y = 8 \cdot 9 = 72 \\ &x \cdot y = 6 \cdot 12 = 72 \end{aligned}$$

\]

Solve for y

\[
y = \frac{72}{x}
\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \frac{x}{72}\)</mcq-option>
<mcq-option>(B) \(y = \frac{x}{18}\)</mcq-option>
<mcq-option>(C) \(y = \frac{18}{x}\)</mcq-option>
<mcq-correct>(D) \(y = \frac{72}{x}\)</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Inverse Variation"
]
}
</post_analysis>