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Answer :
To solve this problem, we want to determine for which temperature the function [tex]\( f(t) = 349.2(0.98)^t \)[/tex] most accurately predicts the cooling time of the oven. We have a table with actual cooling times and temperatures and we need to compare these with the predicted temperatures from the model to find the most accurate prediction.
Here's a step-by-step approach to solving the problem:
1. Identify the Given Data:
We have data for actual oven cooling times and corresponding temperatures:
- Time (minutes) [tex]\( t \)[/tex]: 5, 10, 15, 20, 25
- Actual oven temperatures [tex]\( f(t) \)[/tex]: 315, 285, 260, 235, 210
2. Model Function:
The function given is [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex]. This describes how the temperature of the oven changes over time as it cools.
3. Calculate Predicted Temperatures:
For each time given, plug [tex]\( t \)[/tex] into the function to calculate the predicted temperature.
- For [tex]\( t = 5 \)[/tex]: Calculate [tex]\( f(5) = 349.2 \times (0.98)^5 \)[/tex]
- For [tex]\( t = 10 \)[/tex]: Calculate [tex]\( f(10) = 349.2 \times (0.98)^{10} \)[/tex]
- For [tex]\( t = 15 \)[/tex]: Calculate [tex]\( f(15) = 349.2 \times (0.98)^{15} \)[/tex]
- For [tex]\( t = 20 \)[/tex]: Calculate [tex]\( f(20) = 349.2 \times (0.98)^{20} \)[/tex]
- For [tex]\( t = 25 \)[/tex]: Calculate [tex]\( f(25) = 349.2 \times (0.98)^{25} \)[/tex]
4. Determine the Accuracy:
For each predicted temperature, find the absolute difference between it and the actual temperature to assess accuracy:
- Difference = |Actual Temperature - Predicted Temperature|
5. Identify the Smallest Difference:
Compare the differences calculated for each time. The temperature with the smallest difference between the actual and predicted values indicates the most accurate prediction of the model.
6. Determine the Result:
After the calculations, the temperature that the model predicted most accurately was 285 degrees Fahrenheit. This means the model is most accurate for predicting the temperature of 285 degrees.
Thus, the temperature for which the model most accurately predicts the time spent cooling is 285 degrees Fahrenheit.
Here's a step-by-step approach to solving the problem:
1. Identify the Given Data:
We have data for actual oven cooling times and corresponding temperatures:
- Time (minutes) [tex]\( t \)[/tex]: 5, 10, 15, 20, 25
- Actual oven temperatures [tex]\( f(t) \)[/tex]: 315, 285, 260, 235, 210
2. Model Function:
The function given is [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex]. This describes how the temperature of the oven changes over time as it cools.
3. Calculate Predicted Temperatures:
For each time given, plug [tex]\( t \)[/tex] into the function to calculate the predicted temperature.
- For [tex]\( t = 5 \)[/tex]: Calculate [tex]\( f(5) = 349.2 \times (0.98)^5 \)[/tex]
- For [tex]\( t = 10 \)[/tex]: Calculate [tex]\( f(10) = 349.2 \times (0.98)^{10} \)[/tex]
- For [tex]\( t = 15 \)[/tex]: Calculate [tex]\( f(15) = 349.2 \times (0.98)^{15} \)[/tex]
- For [tex]\( t = 20 \)[/tex]: Calculate [tex]\( f(20) = 349.2 \times (0.98)^{20} \)[/tex]
- For [tex]\( t = 25 \)[/tex]: Calculate [tex]\( f(25) = 349.2 \times (0.98)^{25} \)[/tex]
4. Determine the Accuracy:
For each predicted temperature, find the absolute difference between it and the actual temperature to assess accuracy:
- Difference = |Actual Temperature - Predicted Temperature|
5. Identify the Smallest Difference:
Compare the differences calculated for each time. The temperature with the smallest difference between the actual and predicted values indicates the most accurate prediction of the model.
6. Determine the Result:
After the calculations, the temperature that the model predicted most accurately was 285 degrees Fahrenheit. This means the model is most accurate for predicting the temperature of 285 degrees.
Thus, the temperature for which the model most accurately predicts the time spent cooling is 285 degrees Fahrenheit.
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