Answer :
To determine for which temperature the model most accurately predicts the time spent cooling, we need to compare the predicted temperatures from the function [tex]\( f(t) = 349.2(0.98)^t \)[/tex] with the actual temperatures given in the table.
Here's a step-by-step breakdown of how to find the most accurate prediction:
1. Calculate Predicted Temperatures:
Use the function [tex]\( f(t) = 349.2(0.98)^t \)[/tex] to calculate the predicted temperature for each given time value:
- For [tex]\( t = 5 \)[/tex]
- For [tex]\( t = 10 \)[/tex]
- For [tex]\( t = 15 \)[/tex]
- For [tex]\( t = 20 \)[/tex]
- For [tex]\( t = 25 \)[/tex]
2. Compare Predicted and Actual Temperatures:
Look at the actual temperatures in the table and compare them to the predicted temperatures you calculated.
3. Calculate the Differences:
For each time, calculate the absolute difference between the predicted temperature and the actual temperature.
4. Find the Minimum Difference:
From the differences calculated, identify which time has the smallest difference between the predicted and actual temperatures. This means the model most accurately predicts the temperature at this time.
When you follow these steps, you'll find that the model predicts the temperature most accurately at [tex]\( t = 10 \)[/tex] minutes, with the smallest difference between the predicted and actual temperature. This corresponds to a temperature of 285 degrees Fahrenheit. Thus, the model most accurately predicts the time spent cooling for the temperature 285 degrees Fahrenheit.
Here's a step-by-step breakdown of how to find the most accurate prediction:
1. Calculate Predicted Temperatures:
Use the function [tex]\( f(t) = 349.2(0.98)^t \)[/tex] to calculate the predicted temperature for each given time value:
- For [tex]\( t = 5 \)[/tex]
- For [tex]\( t = 10 \)[/tex]
- For [tex]\( t = 15 \)[/tex]
- For [tex]\( t = 20 \)[/tex]
- For [tex]\( t = 25 \)[/tex]
2. Compare Predicted and Actual Temperatures:
Look at the actual temperatures in the table and compare them to the predicted temperatures you calculated.
3. Calculate the Differences:
For each time, calculate the absolute difference between the predicted temperature and the actual temperature.
4. Find the Minimum Difference:
From the differences calculated, identify which time has the smallest difference between the predicted and actual temperatures. This means the model most accurately predicts the temperature at this time.
When you follow these steps, you'll find that the model predicts the temperature most accurately at [tex]\( t = 10 \)[/tex] minutes, with the smallest difference between the predicted and actual temperature. This corresponds to a temperature of 285 degrees Fahrenheit. Thus, the model most accurately predicts the time spent cooling for the temperature 285 degrees Fahrenheit.
