To determine whether a linear or exponential function better models the population data over time, we can examine how the estimated population changes as the number of years since the last census increases.
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Calculate the differences in population values:
- From year 11 to year 22: \( 43,591 - 36,327 = 7,264 \)
- From year 22 to year 33: \( 36,327 - 29,077 = 7,250 \)
- From year 33 to year 44: \( 29,077 - 21,834 = 7,243 \)
The differences are relatively close, suggesting a linear decrease; however, they are not constant.
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Calculate the ratios of population values:
- From year 11 to year 22: \( \frac{36,327}{43,591} \approx 0.833 \)
- From year 22 to year 33: \( \frac{29,077}{36,327} \approx 0.800 \)
- From year 33 to year 44: \( \frac{21,834}{29,077} \approx 0.750 \)
The ratios are decreasing, suggesting this may hint towards an exponential decay rather than a linear function.
Conclusion:
Based on the changing population values, the differences indicate a consistent decrease, while the ratios highlight a trend typical of exponential decay.
Thus, a more suitable function to model the data would be an exponential function because as \( x \) (the number of years since the last census) increases, the \( y \) (the population) values change in a multiplicative manner (decreasing by a consistent percentage), rather than an additive manner (which would indicate linear change).
In summary, the exponential function would better model the data because as \( x \) increases, the \( y \) values change in a multiplicative way, suggesting a relative rate of decline in the population.