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Answer :
To find the temperature for which the model most accurately predicts the time spent cooling, we should compare the model's predicted temperatures with the actual temperatures given by the table.
First, we calculate the predicted temperature using the model [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex] for each given time:
1. For [tex]\( t = 5 \)[/tex]:
[tex]\[
f(5) = 349.2 \times (0.98)^5 \approx 315.65
\][/tex]
The actual temperature is 315. The difference is:
[tex]\[
|315 - 315.65| \approx 0.65
\][/tex]
2. For [tex]\( t = 10 \)[/tex]:
[tex]\[
f(10) = 349.2 \times (0.98)^{10} \approx 285.32
\][/tex]
The actual temperature is 285. The difference is:
[tex]\[
|285 - 285.32| \approx 0.32
\][/tex]
3. For [tex]\( t = 15 \)[/tex]:
[tex]\[
f(15) = 349.2 \times (0.98)^{15} \approx 257.91
\][/tex]
The actual temperature is 260. The difference is:
[tex]\[
|260 - 257.91| \approx 2.09
\][/tex]
4. For [tex]\( t = 20 \)[/tex]:
[tex]\[
f(20) = 349.2 \times (0.98)^{20} \approx 233.13
\][/tex]
The actual temperature is 235. The difference is:
[tex]\[
|235 - 233.13| \approx 1.87
\][/tex]
5. For [tex]\( t = 25 \)[/tex]:
[tex]\[
f(25) = 349.2 \times (0.98)^{25} \approx 210.73
\][/tex]
The actual temperature is 210. The difference is:
[tex]\[
|210 - 210.73| \approx 0.73
\][/tex]
Now, we compare the differences calculated above to determine which temperature has the smallest absolute difference, indicating the most accurate prediction:
- At [tex]\( t = 5 \)[/tex], the difference is approximately 0.65.
- At [tex]\( t = 10 \)[/tex], the difference is approximately 0.32.
- At [tex]\( t = 15 \)[/tex], the difference is approximately 2.09.
- At [tex]\( t = 20 \)[/tex], the difference is approximately 1.87.
- At [tex]\( t = 25 \)[/tex], the difference is approximately 0.73.
The smallest difference is approximately 0.32, corresponding to the temperature at [tex]\( t = 10 \)[/tex], which is 285 degrees Fahrenheit.
Therefore, the model most accurately predicts the time spent cooling at a temperature of 285 degrees Fahrenheit.
First, we calculate the predicted temperature using the model [tex]\( f(t) = 349.2 \times (0.98)^t \)[/tex] for each given time:
1. For [tex]\( t = 5 \)[/tex]:
[tex]\[
f(5) = 349.2 \times (0.98)^5 \approx 315.65
\][/tex]
The actual temperature is 315. The difference is:
[tex]\[
|315 - 315.65| \approx 0.65
\][/tex]
2. For [tex]\( t = 10 \)[/tex]:
[tex]\[
f(10) = 349.2 \times (0.98)^{10} \approx 285.32
\][/tex]
The actual temperature is 285. The difference is:
[tex]\[
|285 - 285.32| \approx 0.32
\][/tex]
3. For [tex]\( t = 15 \)[/tex]:
[tex]\[
f(15) = 349.2 \times (0.98)^{15} \approx 257.91
\][/tex]
The actual temperature is 260. The difference is:
[tex]\[
|260 - 257.91| \approx 2.09
\][/tex]
4. For [tex]\( t = 20 \)[/tex]:
[tex]\[
f(20) = 349.2 \times (0.98)^{20} \approx 233.13
\][/tex]
The actual temperature is 235. The difference is:
[tex]\[
|235 - 233.13| \approx 1.87
\][/tex]
5. For [tex]\( t = 25 \)[/tex]:
[tex]\[
f(25) = 349.2 \times (0.98)^{25} \approx 210.73
\][/tex]
The actual temperature is 210. The difference is:
[tex]\[
|210 - 210.73| \approx 0.73
\][/tex]
Now, we compare the differences calculated above to determine which temperature has the smallest absolute difference, indicating the most accurate prediction:
- At [tex]\( t = 5 \)[/tex], the difference is approximately 0.65.
- At [tex]\( t = 10 \)[/tex], the difference is approximately 0.32.
- At [tex]\( t = 15 \)[/tex], the difference is approximately 2.09.
- At [tex]\( t = 20 \)[/tex], the difference is approximately 1.87.
- At [tex]\( t = 25 \)[/tex], the difference is approximately 0.73.
The smallest difference is approximately 0.32, corresponding to the temperature at [tex]\( t = 10 \)[/tex], which is 285 degrees Fahrenheit.
Therefore, the model most accurately predicts the time spent cooling at a temperature of 285 degrees Fahrenheit.
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