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To find a model for the data, we can use linear regression, which will allow us to find a mathematical model that describes the decline in the percentage of the rural population in Argentina from 1955 to 2000. The model will have the form:

\[
y = mx + c
\]

where:
- \( y \) is the percentage of the rural population,
- \( x \) is the year,
- \( m \) is the slope of the line (rate of change), and
- \( c \) is the y-intercept (the value of \( y \) when \( x = 0 \)).

Let's convert the years into a form that is easier to work with. We can use \( x = 0 \) to represent the base year (1955), and every subsequent year will be represented as the number of years since 1955. 

Here’s how the data will look:

| Year | Years since 1955 (x) | Percentage rural (y) |
|------|----------------------|----------------------|
| 1955 | 0                    | 30.4                 |
| 1960 | 5                    | 26.4                 |
| 1965 | 10                   | 23.6                 |
| 1970 | 15                   | 21.1                 |
| 1975 | 20                   | 19.0                 |
| 1980 | 25                   | 17.1                 |
| 1985 | 30                   | 15.0                 |
| 1990 | 35                   | 13.0                 |
| 1995 | 40                   | 11.7                 |
| 2000 | 45                   | 10.5                 |

Now, let's perform a linear regression to find the best-fitting line that models this data. I'll calculate the slope \( m \) and intercept \( c \) for the line.The linear regression model for the data is:

\[
y = -0.4305x + 28.467
\]

where:
- \( m = -0.4305 \) (the slope), indicating that the percentage of the rural population decreases by about 0.4305% per year.
- \( c = 28.467 \) (the y-intercept), representing the estimated percentage of the rural population in 1955 (when \( x = 0 \)).

### Estimating the Rural Percentage for 1988 and 2002

Now, to estimate the rural population percentages for 1988 and 2002:
- 1988 is 33 years after 1955, so \( x = 33 \).
- 2002 is 47 years after 1955, so \( x = 47 \).

Let's calculate these values using the model.The estimated rural population percentages are:

- For 1988: **14.26%**
- For 2002: **8.23%**

These values indicate a continued decline in the percentage of Argentina's population living in rural areas based on the linear model.

Would you like to explore more details or have any questions about this process?

### Related Questions
1. How do we interpret the correlation coefficient \( r \) from the linear regression?
2. What would be the impact on the model if we included a quadratic term?
3. How can we use this model to estimate the percentage for a different year, say 2010?
4. What assumptions are we making when using a linear model for this data?
5. Could other types of models, such as exponential decay, be more appropriate for this data?

**Tip:** When working with real-world data, always consider testing multiple models to see which best fits the data trends.

The table shows the percentage of the population of Argentina that has lived in rural areas from 1955 to 2000. Find a model for the data and use it to estimate the rural percentage in 1988 and 2002.

This problem explores the decline in the rural population percentage in Argentina from 1955 to 2000 using linear regression. Learn how to create a mathematical model to estimate the rural population percentage for the years 1988 and 2002. Suitable for students in Grades 9-12.

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