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Sagot :
To determine which model best represents the data set for the temperature of a cup of coffee over time, let’s perform a step-by-step analysis.
### Step-by-Step Solution:
1. Data Extraction:
We have the following pairs of time (minutes) and temperature (degrees Fahrenheit) from the table:
[tex]\[ \text{Time (min)}: [0, 10, 20, 30, 40, 50, 60] \][/tex]
[tex]\[ \text{Temperature (°F)}: [200, 180, 163, 146, 131, 118, 107] \][/tex]
2. Calculate Differences between Consecutive Temperatures:
First, find the difference in temperature between each consecutive time point:
[tex]\[ 200 - 180 = 20 \][/tex]
[tex]\[ 180 - 163 = 17 \][/tex]
[tex]\[ 163 - 146 = 17 \][/tex]
[tex]\[ 146 - 131 = 15 \][/tex]
[tex]\[ 131 - 118 = 13 \][/tex]
[tex]\[ 118 - 107 = 11 \][/tex]
The differences in temperature are:
[tex]\[ [20, 17, 17, 15, 13, 11] \][/tex]
3. Calculate Differences between Consecutive Times:
Next, find the differences in times between each consecutive point:
[tex]\[ 10 - 0 = 10 \][/tex]
[tex]\[ 20 - 10 = 10 \][/tex]
[tex]\[ 30 - 20 = 10 \][/tex]
[tex]\[ 40 - 30 = 10 \][/tex]
[tex]\[ 50 - 40 = 10 \][/tex]
[tex]\[ 60 - 50 = 10 \][/tex]
The differences in time are:
[tex]\[ [10, 10, 10, 10, 10, 10] \][/tex]
4. Calculate Rates of Change:
Then, calculate the rate of change for each time interval:
[tex]\[ \frac{20}{10} = 2 \][/tex]
[tex]\[ \frac{17}{10} = 1.7 \][/tex]
[tex]\[ \frac{17}{10} = 1.7 \][/tex]
[tex]\[ \frac{15}{10} = 1.5 \][/tex]
[tex]\[ \frac{13}{10} = 1.3 \][/tex]
[tex]\[ \frac{11}{10} = 1.1 \][/tex]
The rates of change are:
[tex]\[ [2, 1.7, 1.7, 1.5, 1.3, 1.1] \][/tex]
5. Determine Consistency of Rate of Change:
We need to check if these rates of change are consistent. Notice that the rates of change are not entirely consistent, i.e., they are not all the same.
For a linear model, the rates of change should be roughly constant (additive rate of change). However, even though the rates of change decrease over time, they do so in a pattern that indicates the presence of a multiplicative component (as the changes become smaller progressively).
### Conclusion:
Given that the differences in temperature are decreasing multiplicatively over time, the best model to represent this data set is:
[tex]\[ \textbf{exponential, because there is a relatively consistent multiplicative rate of change} \][/tex]
### Step-by-Step Solution:
1. Data Extraction:
We have the following pairs of time (minutes) and temperature (degrees Fahrenheit) from the table:
[tex]\[ \text{Time (min)}: [0, 10, 20, 30, 40, 50, 60] \][/tex]
[tex]\[ \text{Temperature (°F)}: [200, 180, 163, 146, 131, 118, 107] \][/tex]
2. Calculate Differences between Consecutive Temperatures:
First, find the difference in temperature between each consecutive time point:
[tex]\[ 200 - 180 = 20 \][/tex]
[tex]\[ 180 - 163 = 17 \][/tex]
[tex]\[ 163 - 146 = 17 \][/tex]
[tex]\[ 146 - 131 = 15 \][/tex]
[tex]\[ 131 - 118 = 13 \][/tex]
[tex]\[ 118 - 107 = 11 \][/tex]
The differences in temperature are:
[tex]\[ [20, 17, 17, 15, 13, 11] \][/tex]
3. Calculate Differences between Consecutive Times:
Next, find the differences in times between each consecutive point:
[tex]\[ 10 - 0 = 10 \][/tex]
[tex]\[ 20 - 10 = 10 \][/tex]
[tex]\[ 30 - 20 = 10 \][/tex]
[tex]\[ 40 - 30 = 10 \][/tex]
[tex]\[ 50 - 40 = 10 \][/tex]
[tex]\[ 60 - 50 = 10 \][/tex]
The differences in time are:
[tex]\[ [10, 10, 10, 10, 10, 10] \][/tex]
4. Calculate Rates of Change:
Then, calculate the rate of change for each time interval:
[tex]\[ \frac{20}{10} = 2 \][/tex]
[tex]\[ \frac{17}{10} = 1.7 \][/tex]
[tex]\[ \frac{17}{10} = 1.7 \][/tex]
[tex]\[ \frac{15}{10} = 1.5 \][/tex]
[tex]\[ \frac{13}{10} = 1.3 \][/tex]
[tex]\[ \frac{11}{10} = 1.1 \][/tex]
The rates of change are:
[tex]\[ [2, 1.7, 1.7, 1.5, 1.3, 1.1] \][/tex]
5. Determine Consistency of Rate of Change:
We need to check if these rates of change are consistent. Notice that the rates of change are not entirely consistent, i.e., they are not all the same.
For a linear model, the rates of change should be roughly constant (additive rate of change). However, even though the rates of change decrease over time, they do so in a pattern that indicates the presence of a multiplicative component (as the changes become smaller progressively).
### Conclusion:
Given that the differences in temperature are decreasing multiplicatively over time, the best model to represent this data set is:
[tex]\[ \textbf{exponential, because there is a relatively consistent multiplicative rate of change} \][/tex]
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