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Answer :
- Calculate predicted temperatures using the model for given times.
- Calculate the absolute differences between predicted and observed temperatures.
- Find the minimum absolute difference for each temperature option (0, 100, 300, 400).
- Compare the minimum absolute differences to determine the most accurate temperature: $\boxed{0}$.
### Explanation
1. Calculate Predicted Temperatures
First, we need to calculate the predicted temperatures using the given model $f(t) = 349.2(0.98)^t$ for the times provided in the table: t = 5, 10, 15, 20, and 25 minutes.
2. Predicted Temperatures at Given Times
The predicted temperatures are calculated as follows:
$f(5) = 349.2(0.98)^5
approx 315.65$
$f(10) = 349.2(0.98)^{10}
approx 285.32$
$f(15) = 349.2(0.98)^{15}
approx 257.91$
$f(20) = 349.2(0.98)^{20}
approx 233.13$
$f(25) = 349.2(0.98)^{25}
approx 210.73$
3. Calculate Absolute Differences
Next, we calculate the absolute differences between the predicted temperatures and the observed temperatures from the table:
$|f(5) - 315| = |315.65 - 315| = 0.65$
$|f(10) - 255| = |285.32 - 255| = 30.32$
$|f(15) - 260| = |257.91 - 260| = 2.09$
$|f(20) - 235| = |233.13 - 235| = 1.87$
$|f(25) - 210| = |210.73 - 210| = 0.73$
4. Evaluate Accuracy for Each Temperature Option
Now, we evaluate the accuracy of the model for each of the given temperature options (0, 100, 300, 400) by finding the minimum absolute difference between the predicted temperatures and the observed temperatures.
For temperature option 0, the minimum absolute difference is 0.65 (from time t=5).
To evaluate accuracy for temperature option 100, we calculate the differences between the predicted temperatures and 100:
$|315.65 - 100| = 215.65$
$|285.32 - 100| = 185.32$
$|257.91 - 100| = 157.91$
$|233.13 - 100| = 133.13$
$|210.73 - 100| = 110.73$
The minimum absolute difference for 100 is 110.73.
To evaluate accuracy for temperature option 300, we calculate the differences between the predicted temperatures and 300:
$|315.65 - 300| = 15.65$
$|285.32 - 300| = 14.68$
$|257.91 - 300| = 42.09$
$|233.13 - 300| = 66.87$
$|210.73 - 300| = 89.27$
The minimum absolute difference for 300 is 14.68.
To evaluate accuracy for temperature option 400, we calculate the differences between the predicted temperatures and 400:
$|315.65 - 400| = 84.35$
$|285.32 - 400| = 114.68$
$|257.91 - 400| = 142.09$
$|233.13 - 400| = 166.87$
$|210.73 - 400| = 189.27$
The minimum absolute difference for 400 is 84.35.
5. Compare Minimum Absolute Differences
Comparing the minimum absolute differences for each temperature option, we have:
Minimum absolute difference for 0: 0.65
Minimum absolute difference for 100: 110.73
Minimum absolute difference for 300: 14.68
Minimum absolute difference for 400: 84.35
The temperature option with the smallest minimum absolute difference is 0.
6. Final Answer
Therefore, the model most accurately predicts the time spent cooling for the temperature $\boxed{0}$.
### Examples
Understanding how mathematical models predict real-world phenomena, like oven temperatures, is useful in many fields. For example, engineers use similar models to predict the cooling rates of electronic components to prevent overheating. Chefs might use these models to optimize cooking times and temperatures for consistent results. Even in climate science, models predict temperature changes over time, helping us understand and prepare for climate change.
- Calculate the absolute differences between predicted and observed temperatures.
- Find the minimum absolute difference for each temperature option (0, 100, 300, 400).
- Compare the minimum absolute differences to determine the most accurate temperature: $\boxed{0}$.
### Explanation
1. Calculate Predicted Temperatures
First, we need to calculate the predicted temperatures using the given model $f(t) = 349.2(0.98)^t$ for the times provided in the table: t = 5, 10, 15, 20, and 25 minutes.
2. Predicted Temperatures at Given Times
The predicted temperatures are calculated as follows:
$f(5) = 349.2(0.98)^5
approx 315.65$
$f(10) = 349.2(0.98)^{10}
approx 285.32$
$f(15) = 349.2(0.98)^{15}
approx 257.91$
$f(20) = 349.2(0.98)^{20}
approx 233.13$
$f(25) = 349.2(0.98)^{25}
approx 210.73$
3. Calculate Absolute Differences
Next, we calculate the absolute differences between the predicted temperatures and the observed temperatures from the table:
$|f(5) - 315| = |315.65 - 315| = 0.65$
$|f(10) - 255| = |285.32 - 255| = 30.32$
$|f(15) - 260| = |257.91 - 260| = 2.09$
$|f(20) - 235| = |233.13 - 235| = 1.87$
$|f(25) - 210| = |210.73 - 210| = 0.73$
4. Evaluate Accuracy for Each Temperature Option
Now, we evaluate the accuracy of the model for each of the given temperature options (0, 100, 300, 400) by finding the minimum absolute difference between the predicted temperatures and the observed temperatures.
For temperature option 0, the minimum absolute difference is 0.65 (from time t=5).
To evaluate accuracy for temperature option 100, we calculate the differences between the predicted temperatures and 100:
$|315.65 - 100| = 215.65$
$|285.32 - 100| = 185.32$
$|257.91 - 100| = 157.91$
$|233.13 - 100| = 133.13$
$|210.73 - 100| = 110.73$
The minimum absolute difference for 100 is 110.73.
To evaluate accuracy for temperature option 300, we calculate the differences between the predicted temperatures and 300:
$|315.65 - 300| = 15.65$
$|285.32 - 300| = 14.68$
$|257.91 - 300| = 42.09$
$|233.13 - 300| = 66.87$
$|210.73 - 300| = 89.27$
The minimum absolute difference for 300 is 14.68.
To evaluate accuracy for temperature option 400, we calculate the differences between the predicted temperatures and 400:
$|315.65 - 400| = 84.35$
$|285.32 - 400| = 114.68$
$|257.91 - 400| = 142.09$
$|233.13 - 400| = 166.87$
$|210.73 - 400| = 189.27$
The minimum absolute difference for 400 is 84.35.
5. Compare Minimum Absolute Differences
Comparing the minimum absolute differences for each temperature option, we have:
Minimum absolute difference for 0: 0.65
Minimum absolute difference for 100: 110.73
Minimum absolute difference for 300: 14.68
Minimum absolute difference for 400: 84.35
The temperature option with the smallest minimum absolute difference is 0.
6. Final Answer
Therefore, the model most accurately predicts the time spent cooling for the temperature $\boxed{0}$.
### Examples
Understanding how mathematical models predict real-world phenomena, like oven temperatures, is useful in many fields. For example, engineers use similar models to predict the cooling rates of electronic components to prevent overheating. Chefs might use these models to optimize cooking times and temperatures for consistent results. Even in climate science, models predict temperature changes over time, helping us understand and prepare for climate change.
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