Question

the table represents the temperature of a cup of coffee over time.

temperature of a cup of coffee
\

$$\begin{array}{|c|c|} \\hline \\text{time (minutes)} & \\text{temperature (degrees fahrenheit)} \\\\ \\hline 0 & 200 \\\\ \\hline 10 & 180 \\\\ \\hline 20 & 163 \\\\ \\hline 30 & 146 \\\\ \\hline 40 & 131 \\\\ \\hline 50 & 118 \\\\ \\hline 60 & 107 \\\\ \\hline \\end{array}$$

which model best represents the data set?

• exponential, because there is a relatively consistent multiplicative rate of change
• exponential, because there is a relatively consistent additive rate of change
• linear, because there is a relatively consistent multiplicative rate of change
• linear, because there is a relatively consistent additive rate of change

Explanation:

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

<reasoning>

Analyze successive differences

\[

$$\begin{aligned} &180 - 200 = -20\\ &163 - 180 = -17\\ &146 - 163 = -17\\ &131 - 146 = -15\\ &118 - 131 = -13\\ &107 - 118 = -11 \end{aligned}$$

\]

Analyze successive ratios

\[

$$\begin{aligned} &\frac{180}{200} = 0.900\\ &\frac{163}{180} \approx 0.906\\ &\frac{146}{163} \approx 0.896\\ &\frac{131}{146} \approx 0.897\\ &\frac{118}{131} \approx 0.901\\ &\frac{107}{118} \approx 0.907 \end{aligned}$$

\]

Determine the best model

\[

$$\begin{aligned} &\text{Ratios are extremely consistent } (\approx 0.90).\\ &\text{Multiplicative rate of change indicates an exponential model.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>exponential, because there is a relatively consistent multiplicative rate of change</mcq-correct>
<mcq-option>exponential, because there is a relatively consistent additive rate of change</mcq-option>
<mcq-option>linear, because there is a relatively consistent multiplicative rate of change</mcq-option>
<mcq-option>linear, because there is a relatively consistent additive rate of change</mcq-option>
</answer>

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

Answer:

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

<reasoning>

Analyze successive differences

\[

$$\begin{aligned} &180 - 200 = -20\\ &163 - 180 = -17\\ &146 - 163 = -17\\ &131 - 146 = -15\\ &118 - 131 = -13\\ &107 - 118 = -11 \end{aligned}$$

\]

Analyze successive ratios

\[

$$\begin{aligned} &\frac{180}{200} = 0.900\\ &\frac{163}{180} \approx 0.906\\ &\frac{146}{163} \approx 0.896\\ &\frac{131}{146} \approx 0.897\\ &\frac{118}{131} \approx 0.901\\ &\frac{107}{118} \approx 0.907 \end{aligned}$$

\]

Determine the best model

\[

$$\begin{aligned} &\text{Ratios are extremely consistent } (\approx 0.90).\\ &\text{Multiplicative rate of change indicates an exponential model.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>exponential, because there is a relatively consistent multiplicative rate of change</mcq-correct>
<mcq-option>exponential, because there is a relatively consistent additive rate of change</mcq-option>
<mcq-option>linear, because there is a relatively consistent multiplicative rate of change</mcq-option>
<mcq-option>linear, because there is a relatively consistent additive rate of change</mcq-option>
</answer>

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