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Answer :
To determine which temperature the model most accurately predicts the time spent cooling, we need to compare the temperatures predicted by the model with the actual temperatures for different times. The function given is [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex].
Here’s a step-by-step solution:
1. Calculate Model Predictions:
For each of the given times (5, 10, 15, 20, 25 minutes), calculate the temperature using the model function [tex]\( f(t) \)[/tex].
- At 5 minutes: [tex]\( f(5) = 349.2 \times (0.98)^5 \approx 315.65 \)[/tex]
- At 10 minutes: [tex]\( f(10) = 349.2 \times (0.98)^{10} \approx 285.32 \)[/tex]
- At 15 minutes: [tex]\( f(15) = 349.2 \times (0.98)^{15} \approx 257.91 \)[/tex]
- At 20 minutes: [tex]\( f(20) = 349.2 \times (0.98)^{20} \approx 233.13 \)[/tex]
- At 25 minutes: [tex]\( f(25) = 349.2 \times (0.98)^{25} \approx 210.73 \)[/tex]
2. Compare Model Predictions with Actual Temperatures:
Calculate the differences between the temperatures predicted by the model and the actual temperatures given in the table.
- At 5 minutes: [tex]\(|315.65 - 315| \approx 0.65\)[/tex]
- At 10 minutes: [tex]\(|285.32 - 285| \approx 0.32\)[/tex]
- At 15 minutes: [tex]\(|257.91 - 260| \approx 2.09\)[/tex]
- At 20 minutes: [tex]\(|233.13 - 235| \approx 1.87\)[/tex]
- At 25 minutes: [tex]\(|210.73 - 210| \approx 0.73\)[/tex]
3. Find the Smallest Difference:
Identify which time corresponds to the smallest difference between the predicted and actual temperature. The smallest difference is approximately 0.32, occurring at 10 minutes.
4. Determine the Predicted Temperature:
The predicted temperature where the model is most accurate is 300 degrees Fahrenheit.
Therefore, the temperature for which the model most accurately predicts the time spent cooling is 300 degrees Fahrenheit.
Here’s a step-by-step solution:
1. Calculate Model Predictions:
For each of the given times (5, 10, 15, 20, 25 minutes), calculate the temperature using the model function [tex]\( f(t) \)[/tex].
- At 5 minutes: [tex]\( f(5) = 349.2 \times (0.98)^5 \approx 315.65 \)[/tex]
- At 10 minutes: [tex]\( f(10) = 349.2 \times (0.98)^{10} \approx 285.32 \)[/tex]
- At 15 minutes: [tex]\( f(15) = 349.2 \times (0.98)^{15} \approx 257.91 \)[/tex]
- At 20 minutes: [tex]\( f(20) = 349.2 \times (0.98)^{20} \approx 233.13 \)[/tex]
- At 25 minutes: [tex]\( f(25) = 349.2 \times (0.98)^{25} \approx 210.73 \)[/tex]
2. Compare Model Predictions with Actual Temperatures:
Calculate the differences between the temperatures predicted by the model and the actual temperatures given in the table.
- At 5 minutes: [tex]\(|315.65 - 315| \approx 0.65\)[/tex]
- At 10 minutes: [tex]\(|285.32 - 285| \approx 0.32\)[/tex]
- At 15 minutes: [tex]\(|257.91 - 260| \approx 2.09\)[/tex]
- At 20 minutes: [tex]\(|233.13 - 235| \approx 1.87\)[/tex]
- At 25 minutes: [tex]\(|210.73 - 210| \approx 0.73\)[/tex]
3. Find the Smallest Difference:
Identify which time corresponds to the smallest difference between the predicted and actual temperature. The smallest difference is approximately 0.32, occurring at 10 minutes.
4. Determine the Predicted Temperature:
The predicted temperature where the model is most accurate is 300 degrees Fahrenheit.
Therefore, the temperature for which the model most accurately predicts the time spent cooling is 300 degrees Fahrenheit.
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