Question
The estimated population of a certain city over time is given in the table below. Answer the questions below to determine what kind of function would best fit the data, linear or exponential.
Number of Years Since Last Census, x 11 22 33 44
Estimated Population, f(x) 46, comma, 53846,538 44, comma, 72144,721 43, comma, 19143,191 41, comma, 29841,298
Answer
Attempt 1 out of 2
function would best fit the data because as xx increases, the yy values change
. The
of this function is approximately
.
Number of Years Since Last Census, x 11 22 33 44
Estimated Population, f(x) 46, comma, 53846,538 44, comma, 72144,721 43, comma, 19143,191 41, comma, 29841,298
Answer
Attempt 1 out of 2
function would best fit the data because as xx increases, the yy values change
. The
of this function is approximately
.
Answers
GPT-4o mini
To determine whether a linear or exponential function fits the data best, we should analyze how the population changes as time progresses.
1. **Linear Function Check**: A linear function would show a constant rate of change in population over time. We can find the differences between consecutive population estimates.
- From year 11 to year 22: 44,721 - 46,538 = -1,817
- From year 22 to year 33: 43,191 - 44,721 = -1,530
- From year 33 to year 44: 41,298 - 43,191 = -1,893
As seen, the differences are not constant, therefore, this does not fit a linear function.
2. **Exponential Function Check**: An exponential function would demonstrate that the population changes at a constant percentage rate. We can find the ratios of the population between the years.
Calculate the growth factor between each pair of years:
- From year 11 to year 22: \( \frac{44,721}{46,538} \approx 0.96 \)
- From year 22 to year 33: \( \frac{43,191}{44,721} \approx 0.97 \)
- From year 33 to year 44: \( \frac{41,298}{43,191} \approx 0.95 \)
The ratios are close to 1 but less than 1, suggesting a consistent decrease, which indicates that it could fit an exponential decay model.
Based on these observations, we can conclude that:
The **exponential function** would best fit the data because as \(x\) increases, the \(y\) values change at a decreasing rate. The **decay rate** of this function is approximately around 3-5% per year, as indicated by the ratios calculated.
So, your answer can be filled in as follows:
The **exponential function** would best fit the data because as \(x\) increases, the \(y\) values change **at a decreasing rate**. The **decay rate** of this function is approximately **3-5%**.
1. **Linear Function Check**: A linear function would show a constant rate of change in population over time. We can find the differences between consecutive population estimates.
- From year 11 to year 22: 44,721 - 46,538 = -1,817
- From year 22 to year 33: 43,191 - 44,721 = -1,530
- From year 33 to year 44: 41,298 - 43,191 = -1,893
As seen, the differences are not constant, therefore, this does not fit a linear function.
2. **Exponential Function Check**: An exponential function would demonstrate that the population changes at a constant percentage rate. We can find the ratios of the population between the years.
Calculate the growth factor between each pair of years:
- From year 11 to year 22: \( \frac{44,721}{46,538} \approx 0.96 \)
- From year 22 to year 33: \( \frac{43,191}{44,721} \approx 0.97 \)
- From year 33 to year 44: \( \frac{41,298}{43,191} \approx 0.95 \)
The ratios are close to 1 but less than 1, suggesting a consistent decrease, which indicates that it could fit an exponential decay model.
Based on these observations, we can conclude that:
The **exponential function** would best fit the data because as \(x\) increases, the \(y\) values change at a decreasing rate. The **decay rate** of this function is approximately around 3-5% per year, as indicated by the ratios calculated.
So, your answer can be filled in as follows:
The **exponential function** would best fit the data because as \(x\) increases, the \(y\) values change **at a decreasing rate**. The **decay rate** of this function is approximately **3-5%**.