Answer :
To solve this problem, we need to determine which of the given temperatures—0, 100, 300, or 400—can be predicted with the highest accuracy by the model [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex] based on the provided data.
### Step-by-Step Solution:
1. Understand the Model:
[tex]\[
f(t) = 349.2 \times (0.98)^t
\][/tex]
This equation models how the temperature of an oven decreases over time [tex]\( t \)[/tex] in minutes.
2. Examine Given Data:
The provided table gives the actual temperatures at specific times:
- At 5 minutes, the temperature is 315°F.
- At 10 minutes, the temperature is 285°F.
- At 15 minutes, the temperature is 260°F.
- At 20 minutes, the temperature is 235°F.
- At 25 minutes, the temperature is 210°F.
3. Identify the Target Temperatures:
We need to assess how closely the model predicts the following temperatures: 0, 100, 300, 400°F.
4. Calculate Model Temperatures and Find Errors:
For each target temperature, we want to determine how well our model can approximate these using the times from the data table.
5. Determine the Best Fit Temperature:
By comparing how close each target temperature can be predicted by the model given the actual data temperatures, we find that the model most accurately predicts a temperature of 0°F. The error associated with predicting a temperature of 0°F is the smallest among all the given options.
Hence, for the given model and data, the temperature for which the model most accurately predicts the time spent cooling is 0°F.
### Step-by-Step Solution:
1. Understand the Model:
[tex]\[
f(t) = 349.2 \times (0.98)^t
\][/tex]
This equation models how the temperature of an oven decreases over time [tex]\( t \)[/tex] in minutes.
2. Examine Given Data:
The provided table gives the actual temperatures at specific times:
- At 5 minutes, the temperature is 315°F.
- At 10 minutes, the temperature is 285°F.
- At 15 minutes, the temperature is 260°F.
- At 20 minutes, the temperature is 235°F.
- At 25 minutes, the temperature is 210°F.
3. Identify the Target Temperatures:
We need to assess how closely the model predicts the following temperatures: 0, 100, 300, 400°F.
4. Calculate Model Temperatures and Find Errors:
For each target temperature, we want to determine how well our model can approximate these using the times from the data table.
5. Determine the Best Fit Temperature:
By comparing how close each target temperature can be predicted by the model given the actual data temperatures, we find that the model most accurately predicts a temperature of 0°F. The error associated with predicting a temperature of 0°F is the smallest among all the given options.
Hence, for the given model and data, the temperature for which the model most accurately predicts the time spent cooling is 0°F.
