Question

the function (f(t) = 349.2(0.98)^t) models the relationship between (t), the time an oven spends cooling and the temperature of the oven.

oven cooling time

time (minutes) (t)	oven temperature (degrees fahrenheit) (f(t))
10	285
15	260
20	235
25	210

for which temperature will the model most accurately predict the time spent cooling?

(a) 0
(b) 100
(c) 300
(d) 400

Explanation:

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"concepts_used": [
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</pre_analysis>

<reasoning>

Analyze the given model and data range

Using the Exponential Modeling knowledge point
\[

$$\begin{aligned} &f(t) = 349.2(0.98)^t\\ &\text{Data domain } t \in [5, 25]\\ &\text{Data range } f(t) \in [210, 315] \end{aligned}$$

\]

Evaluate interpolation versus extrapolation

Using the Interpolation and Extrapolation knowledge points
\[

$$\begin{aligned} &T = 300 \in [210, 315] \implies \text{Interpolation}\\ &T \in \{0, 100, 400\} otin [210, 315] \implies \text{Extrapolation} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>0</mcq-option>
<mcq-option>100</mcq-option>
<mcq-correct>300</mcq-correct>
<mcq-option>400</mcq-option>
</answer>

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

Answer:

<pre_analysis>
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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
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"Interpolation",
"Extrapolation",
"Domain and Range"
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}
</pre_analysis>

<reasoning>

Analyze the given model and data range

Using the Exponential Modeling knowledge point
\[

$$\begin{aligned} &f(t) = 349.2(0.98)^t\\ &\text{Data domain } t \in [5, 25]\\ &\text{Data range } f(t) \in [210, 315] \end{aligned}$$

\]

Evaluate interpolation versus extrapolation

Using the Interpolation and Extrapolation knowledge points
\[

$$\begin{aligned} &T = 300 \in [210, 315] \implies \text{Interpolation}\\ &T \in \{0, 100, 400\} otin [210, 315] \implies \text{Extrapolation} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>0</mcq-option>
<mcq-option>100</mcq-option>
<mcq-correct>300</mcq-correct>
<mcq-option>400</mcq-option>
</answer>

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