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Answer :
To determine which temperature the model most accurately predicts based on the given function and data, we can follow these steps:
1. Understand the Function: The function given is [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex]. This represents the temperature of the oven depending on the time [tex]\( t \)[/tex] in minutes.
2. Identify Actual Data: We have a table of actual temperatures recorded at certain times:
- At 5 minutes: 315°F
- At 10 minutes: 285°F
- At 15 minutes: 260°F
- At 20 minutes: 235°F
- At 25 minutes: 210°F
3. Predict the Temperatures Using the Model: Use the function to predict the temperature at each of these times:
- For [tex]\( t = 5 \)[/tex], calculate [tex]\( f(5) \)[/tex]
- For [tex]\( t = 10 \)[/tex], calculate [tex]\( f(10) \)[/tex]
- For [tex]\( t = 15 \)[/tex], calculate [tex]\( f(15) \)[/tex]
- For [tex]\( t = 20 \)[/tex], calculate [tex]\( f(20) \)[/tex]
- For [tex]\( t = 25 \)[/tex], calculate [tex]\( f(25) \)[/tex]
4. Compare with Actual Data: For each time point, compare the predicted temperature from the model with the actual temperature recorded in the table. Calculate the absolute difference between the predicted and actual temperatures.
5. Identify the Best Fit: Determine which time has the smallest difference between the predicted temperature and the actual recorded temperature. This will show us for which temperature the model is most accurate.
After comparing the differences for each time point, the temperature of 285°F at 10 minutes has the smallest difference between the predicted and actual temperatures. Therefore, the model most accurately predicts the time spent cooling for the temperature of 285°F.
1. Understand the Function: The function given is [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex]. This represents the temperature of the oven depending on the time [tex]\( t \)[/tex] in minutes.
2. Identify Actual Data: We have a table of actual temperatures recorded at certain times:
- At 5 minutes: 315°F
- At 10 minutes: 285°F
- At 15 minutes: 260°F
- At 20 minutes: 235°F
- At 25 minutes: 210°F
3. Predict the Temperatures Using the Model: Use the function to predict the temperature at each of these times:
- For [tex]\( t = 5 \)[/tex], calculate [tex]\( f(5) \)[/tex]
- For [tex]\( t = 10 \)[/tex], calculate [tex]\( f(10) \)[/tex]
- For [tex]\( t = 15 \)[/tex], calculate [tex]\( f(15) \)[/tex]
- For [tex]\( t = 20 \)[/tex], calculate [tex]\( f(20) \)[/tex]
- For [tex]\( t = 25 \)[/tex], calculate [tex]\( f(25) \)[/tex]
4. Compare with Actual Data: For each time point, compare the predicted temperature from the model with the actual temperature recorded in the table. Calculate the absolute difference between the predicted and actual temperatures.
5. Identify the Best Fit: Determine which time has the smallest difference between the predicted temperature and the actual recorded temperature. This will show us for which temperature the model is most accurate.
After comparing the differences for each time point, the temperature of 285°F at 10 minutes has the smallest difference between the predicted and actual temperatures. Therefore, the model most accurately predicts the time spent cooling for the temperature of 285°F.
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