To determine the temperature at which the model [tex]\( f(t) = 349.2 \cdot (0.98)^t \)[/tex] predicts the cooling time most accurately, we can compare the given temperatures in the table with the model predictions and calculate the mean squared error (MSE) for each given model temperature [tex]\( f(t) \)[/tex].
Given data from the table:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time (minutes)} & \text{Oven temperature (°F)} \\
\hline
5 & 315 \\
\hline
10 & 285 \\
\hline
15 & 260 \\
\hline
20 & 235 \\
\hline
25 & 210 \\
\hline
\end{array}
\][/tex]
We have four specific model temperatures to consider: [tex]\( f(0) \)[/tex], [tex]\( f(100) \)[/tex], [tex]\( f(300) \)[/tex], and [tex]\( f(400) \)[/tex].
1. Calculating model temperatures:
[tex]\[
\begin{aligned}
f(0) &= 349.2 \cdot (0.98)^0 = 349.2, \\
f(100) &= 349.2 \cdot (0.98)^{100}, \\
f(300) &= 349.2 \cdot (0.98)^{300}, \\
f(400) &= 349.2 \cdot (0.98)^{400}.
\end{aligned}
\][/tex]
2. Calculating the mean squared error for each model temperature and given data:
The mean squared error (MSE) for each model temperature is calculated as follows:
[tex]\[
\text{MSE} = \frac{1}{N} \sum_{i=1}^{N} (T_i - T_{\text{model}})^2,
\][/tex]
where [tex]\( T_i \)[/tex] are the observed temperatures from the given data ([tex]\( 315, 285, 260, 235, 210 \)[/tex]) and [tex]\( T_{\text{model}} \)[/tex] is the model temperature ([tex]\( f(0) \)[/tex], [tex]\( f(100) \)[/tex], [tex]\( f(300) \)[/tex], [tex]\( f(400) \)[/tex]).
3. Comparison and finding the minimal MSE:
After calculating the MSE for each model temperature, the results are compared to find the one with the minimum MSE, which indicates the model temperature that predicts the cooling time most accurately.
Based on the given question and the calculated result:
[tex]\[
\text{Minimum Mean Squared Error} = 9133.24, \quad \text{Temperature Index} = 1.
\][/tex]
This means that the model temperature that minimizes the error is [tex]\( f(0) = 349.2 \)[/tex].
Therefore, the model [tex]\( f(t) = 349.2 \cdot (0.98)^t \)[/tex] predicts the cooling time most accurately when the initial temperature is:
[tex]\[
\boxed{0 \text{ degrees Fahrenheit}}
\][/tex]
The detailed step-by-step solution shows that the temperature for which the model most accurately predicts the time spent cooling is indeed [tex]\( 0 \text{ degrees Fahrenheit} \)[/tex].
