Question

the table represents the temperature of a cup of coffee over time.
temperature of a cup of coffee

time (minutes)	temperature (degrees fahrenheit)
10	180
20	163
30	146
40	131
50	118
60	107

which model best represents the data set?

• exponential, because there is a relatively consistent multiplicative rate of change
• exponential, because there is a relatively consistent additive rate of change
• linear, because there is a relatively consistent multiplicative rate of change
• linear, because there is a relatively consistent additive rate of change

Explanation:

Step1: Analyze Additive Change

Calculate the difference (additive change) between consecutive temperatures:
$200 - 180 = 20$, $180 - 163 = 17$, $163 - 146 = 17$, $146 - 131 = 15$, $131 - 118 = 13$, $118 - 107 = 11$.
These differences are not consistent, so not linear (linear needs consistent additive change).

Step2: Analyze Multiplicative Change

Calculate the ratio (multiplicative change) between consecutive temperatures:
$\frac{180}{200} = 0.9$, $\frac{163}{180} \approx 0.906$, $\frac{146}{163} \approx 0.896$, $\frac{131}{146} \approx 0.897$, $\frac{118}{131} \approx 0.901$, $\frac{107}{118} \approx 0.907$.
These ratios are relatively consistent (around 0.9 - 0.91), indicating a multiplicative rate of change, characteristic of exponential models.

Step3: Eliminate Incorrect Options

• Linear models require consistent additive (not multiplicative) change, so eliminate linear options.
• Exponential models use multiplicative (not additive) rate of change, so the second exponential option is wrong.

Answer:

exponential, because there is a relatively consistent multiplicative rate of change