To determine if the temperatures in Jacob's city increased linearly or exponentially over the five months, let's analyze the given temperature data step-by-step.
Step 1: Calculate the differences between consecutive months
We first find the differences between temperatures of consecutive months:
- Difference between Month 2 and Month 1: [tex]\( 7.2 - 5 = 2.2 \)[/tex]
- Difference between Month 3 and Month 2: [tex]\( 9.4 - 7.2 = 2.2 \)[/tex]
- Difference between Month 4 and Month 3: [tex]\( 11.6 - 9.4 = 2.2 \)[/tex]
- Difference between Month 5 and Month 4: [tex]\( 13.8 - 11.6 = 2.2 \)[/tex]
The differences between consecutive months are: [tex]\([2.2, 2.2, 2.2, 2.2]\)[/tex]
Since the differences are consistent (all equal to 2.2), this suggests a linear increase because the temperature increased by the same amount each month.
Step 2: Calculate the ratios between consecutive months
Next, we find the ratios to check for an exponential increase:
- Ratio between Month 2 and Month 1: [tex]\( \frac{7.2}{5} \approx 1.44 \)[/tex]
- Ratio between Month 3 and Month 2: [tex]\( \frac{9.4}{7.2} \approx 1.3056 \)[/tex]
- Ratio between Month 4 and Month 3: [tex]\( \frac{11.6}{9.4} \approx 1.2340 \)[/tex]
- Ratio between Month 5 and Month 4: [tex]\( \frac{13.8}{11.6} \approx 1.1897 \)[/tex]
The ratios between consecutive months are: [tex]\([1.44, 1.3056, 1.2340, 1.1897]\)[/tex]
Since the ratios are not consistent, this does not support an exponential increase. An exponential increase would show a constant percentage increase (i.e., the same ratio) each month.
Step 3: Conclusion
Given that the temperature increased by the same amount each month (consistent differences) and the ratios between consecutive months are not constant, we can conclude that the temperature in Jacob's city increased linearly over the given period.
Answer:
Linearly, because the table shows that the temperature increased by the same amount each month.
