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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)

\[
\begin{array}{|c|c|}
\hline
\text{Years Since 1970} (x) & \text{Population (in thousands)} (y) \\
\hline
0 & 489 \\
10 & 801 \\
20 & 1,202 \\
30 & 1,998 \\
40 & 2,959 \\
\hline
\end{array}
\]

### 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).

---

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.

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.

Learn how to find the best-fit quadratic model for population growth data and use it to predict future populations. This example uses data from 1970 and calculates the estimated population for 2020 to the nearest tenth.

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