Question

the table shows the estimated number of lines of code written by computer programmers per hour when \\(x\\) people are working.

productivity
\

$$\begin{tabular}{|c|c|} \\hline people working & lines of code written hourly \\\\ \\hline 2 & 50 \\\\ \\hline 4 & 110 \\\\ \\hline 6 & 160 \\\\ \\hline 8 & 210 \\\\ \\hline 10 & 270 \\\\ \\hline 12 & 320 \\\\ \\hline \\end{tabular}$$

which model best represents the data?

• \\(y = 26.9x - 1.3\\)
• \\(y = 27x - 4\\)
• \\(y = 47(1.191)^x\\)
• \\(y = 34(1.204)^x\\)

Explanation:

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"new_concepts": [],
"current_concepts": [
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}
</pre_analysis>

<reasoning>

Analyze the rate of change in the data

\[

$$\begin{aligned} &\Delta x = 4 - 2 = 6 - 4 = 8 - 6 = 10 - 8 = 12 - 10 = 2 \\ &\Delta y_1 = 110 - 50 = 60 \\ &\Delta y_2 = 160 - 110 = 50 \\ &\Delta y_3 = 210 - 160 = 50 \\ &\Delta y_4 = 270 - 210 = 60 \\ &\Delta y_5 = 320 - 270 = 50 \\ &\text{Average } \Delta y \approx 54 \\ &m = \frac{\Delta y}{\Delta x} \approx \frac{54}{2} = 27 \end{aligned}$$

\]

Determine the linear model equation

\[

$$\begin{aligned} &y - y_1 = m(x - x_1) \\ &y - 50 = 27(x - 2) \\ &y - 50 = 27x - 54 \\ &y = 27x - 4 \end{aligned}$$

\]

Verify the model against data points

\[

$$\begin{aligned} &x = 2 \implies y = 27(2) - 4 = 50 \\ &x = 6 \implies y = 27(6) - 4 = 158 \approx 160 \\ &x = 12 \implies y = 27(12) - 4 = 320 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = 26.9x - 1.3\)</mcq-option>
<mcq-correct>(B) \(y = 27x - 4\)</mcq-correct>
<mcq-option>(C) \(y = 47(1.191)^x\)</mcq-option>
<mcq-option>(D) \(y = 34(1.204)^x\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Modeling"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
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"concepts_used": [
"Function Modeling",
"Linear Modeling",
"Constant Rate of Change"
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"new_concepts": [],
"current_concepts": [
"Function Modeling",
"Linear Modeling",
"Constant Rate of Change"
]
}
</pre_analysis>

<reasoning>

Analyze the rate of change in the data

\[

$$\begin{aligned} &\Delta x = 4 - 2 = 6 - 4 = 8 - 6 = 10 - 8 = 12 - 10 = 2 \\ &\Delta y_1 = 110 - 50 = 60 \\ &\Delta y_2 = 160 - 110 = 50 \\ &\Delta y_3 = 210 - 160 = 50 \\ &\Delta y_4 = 270 - 210 = 60 \\ &\Delta y_5 = 320 - 270 = 50 \\ &\text{Average } \Delta y \approx 54 \\ &m = \frac{\Delta y}{\Delta x} \approx \frac{54}{2} = 27 \end{aligned}$$

\]

Determine the linear model equation

\[

$$\begin{aligned} &y - y_1 = m(x - x_1) \\ &y - 50 = 27(x - 2) \\ &y - 50 = 27x - 54 \\ &y = 27x - 4 \end{aligned}$$

\]

Verify the model against data points

\[

$$\begin{aligned} &x = 2 \implies y = 27(2) - 4 = 50 \\ &x = 6 \implies y = 27(6) - 4 = 158 \approx 160 \\ &x = 12 \implies y = 27(12) - 4 = 320 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = 26.9x - 1.3\)</mcq-option>
<mcq-correct>(B) \(y = 27x - 4\)</mcq-correct>
<mcq-option>(C) \(y = 47(1.191)^x\)</mcq-option>
<mcq-option>(D) \(y = 34(1.204)^x\)</mcq-option>
</answer>

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