This table shows the population of a city every 10 years since 1970. Write the best-fit model for this data. Use this model to estimate the population in 2020, to the nearest tenth.
Solution
The table provided shows the population of a city every 10 years since 1970, and the question asks to write the best-fit model and estimate the population in 2020. Here's how we can approach this:
Data:
x: Years since 1970
y: Population (in thousands)
\hline
\text{Years Since 1970} (x) & \text{Population (in thousands)} (y) \\
\hline
0 & 489 \\
10 & 801 \\
20 & 1,202 \\
30 & 1,998 \\
40 & 2,959 \\
\hline
\end{array}$$
### Best-fit Model
The model is quadratic, and we are given options that include quadratic functions like:
1. $$f(x) = 1.209x^2 + 12.999x + 504.257$$
2. $$f(x) = 1.209x^2 + 13.000x + 504.257$$
We'll estimate the population in 2020, which is 50 years after 1970, so $$x = 50$$.
### Calculating for $$x = 50$$:
Substitute $$x = 50$$ into both models:
#### First Model:
$$f(x) = 1.209(50)^2 + 12.999(50) + 504.257$$
$$f(50) = 1.209(2500) + 12.999(50) + 504.257$$
$$f(50) = 3022.5 + 649.95 + 504.257 = 4176.707 \text{ thousand}$$
#### Second Model:
$$f(x) = 1.209(50)^2 + 13.000(50) + 504.257$$
$$f(50) = 1.209(2500) + 13.000(50) + 504.257$$
$$f(50) = 3022.5 + 650 + 504.257 = 4176.757 \text{ thousand}$$
Both models give approximately the same result: **4176.7 thousand** (or **4.177 million** to the nearest tenth).
### Conclusion:
The estimated population in 2020, using the best-fit quadratic model, is **4176.7 thousand** (option 1 is the correct answer).
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Would you like more details or further clarification?
### Related Questions:
1. How is a quadratic function derived from data points like this?
2. Why is a quadratic model better than a linear one for this data?
3. How do changes in coefficients affect the shape of a quadratic model?
4. What would the population be in 2030 using this model?
5. How could we improve the accuracy of the population prediction?
#### Tip:
For quadratic models, the rate of change increases over time, reflecting accelerating growth in population estimates.
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