Question

The estimated population of a certain city over time is given in the table below. Answer the questions below to explain what kind of function would better model the data, linear or exponential.
Number of Years Since Last Census, x 11 22 33 44
Estimated Population, f(x) 60, comma, 35660,356 70, comma, 28670,286 80, comma, 26680,266 90, comma, 24690,246
(a linear or an exponential) function would better model the data because as x increases, the y values change (additively or multiplicatively) The (common difference/slope or common ratio/multiplier/base) of this function is approximately (???)

Answer 1

To determine whether a linear or exponential function better models the given population data, we need to analyze how the estimated population changes with respect to time (years since the last census).

### Data Given:
- Years Since Last Census (x): 11, 22, 33, 44
- Estimated Population (f(x)): 60,356; 70,286; 80,266; 90,246

### Step 1: Check for Linear Behavior
If the function is linear, the change in population should be constant with each increment of time. We will check the differences in population between consecutive data points.

- From 11 to 22 years:
- Population change = 70,286 - 60,356 = 9,930
- From 22 to 33 years:
- Population change = 80,266 - 70,286 = 9,980
- From 33 to 44 years:
- Population change = 90,246 - 80,266 = 9,980

The change in population is approximately the same, suggesting a linear trend, but we also notice that the increases are slightly different.

### Step 2: Check for Exponential Behavior
If the function is exponential, we can examine the ratios of the population values in relation to one another, which should be constant in a multiplicative manner.

- Ratio from 11 to 22 years:
- Ratio = 70,286 / 60,356 ≈ 1.162
- Ratio from 22 to 33 years:
- Ratio = 80,266 / 70,286 ≈ 1.142
- Ratio from 33 to 44 years:
- Ratio = 90,246 / 80,266 ≈ 1.122

The ratios are not constant, indicating that it does not fit an exponential model either.

### Conclusion:
Overall, the differences indicate a predominantly linear trend, as the population changes additively. Therefore, we can conclude:

**A linear function would better model the data because as x increases, the y values change additively. The common difference (slope) of this function is approximately 9,980.**

>Note: A more rigorous statistical analysis, like calculating R² values or fitting both linear and exponential models, could further refine this conclusion. However, based on the provided changes and ratios, a linear function appears appropriate.