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: Check additive rate of change

For linear models, the additive rate of change (difference) should be relatively consistent.

• From \(t = 0\) to \(t=10\): \(200 - 180=20\)
• From \(t = 10\) to \(t = 20\): \(180-163 = 17\)
• From \(t = 20\) to \(t=30\): \(163 - 146=17\)
• From \(t = 30\) to \(t = 40\): \(146-131 = 15\)
• From \(t = 40\) to \(t=50\): \(131 - 118=13\)
• From \(t = 50\) to \(t = 60\): \(118-107 = 11\)

The additive rate of change is not consistent, so it is not a linear model.

Step2: Check multiplicative rate of change

For exponential models, the multiplicative rate of change (ratio) should be relatively consistent.

• \(\frac{180}{200}=0.9\)
• \(\frac{163}{180}\approx0.906\)
• \(\frac{146}{163}\approx0.896\)
• \(\frac{131}{146}\approx0.897\)
• \(\frac{118}{131}\approx0.901\)
• \(\frac{107}{118}\approx0.907\)

The multiplicative rate of change is relatively consistent.

Answer:

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