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Based on the image, we are given a table of personal income per capita for Georgia in constant 2005 dollars over several years, along with a scatter plot and the question of whether a linear model fits the data.

To address the questions:

1. **Linear Model Fit**:  
   Looking at the scatter plot, the blue points closely follow the red line. This indicates that a linear model fits the data well.

2. **Linear Model Equation**:  
   We need to derive the equation for the linear model $ I(t) $, where $ I(t) $ represents the per capita income in year $ t $ since 1980.

   From the data points, it seems the relationship is linear, and we can estimate the slope (rate of change) and intercept (starting value). Using two points from the table:
   - Year 1 (1981): Income = 9393
   - Year 26 (2006): Income = 32095

   The slope $ m $ is calculated as:
   \[
   m = \frac{32095 - 9393}{26 - 1} = \frac{22702}{25} = 908.08
   \]
   Now, using the slope-intercept form of the line $ I(t) = mt + b $, where $ b $ is the intercept, substitute $ t = 1 $ and $ I(1) = 9393 $:
   \[
   9393 = 908.08(1) + b
   \]
   \[
   b = 9393 - 908.08 = 8484.92
   \]
   Therefore, the linear model is approximately:
   \[
   I(t) = 908.08t + 8484.92
   \]

3. **Per Capita Income in 2015**:  
   Since 2015 corresponds to $ t = 2015 - 1980 = 35 $, substitute $ t = 35 $ into the model:
   \[
   I(35) = 908.08(35) + 8484.92 = 31782.8 + 8484.92 = 40267.72
   \]
   Rounded to the nearest dollar, the per capita income in 2015 is approximately **$40,268**.

4. **Year in which Income will be $38,200**:  
   To find the year when income will be $38,200, solve for $ t $ in the equation $ 38200 = 908.08t + 8484.92 $:
   \[
   38200 = 908.08t + 8484.92
   \]
   \[
   29715.08 = 908.08t
   \]
   \[
   t = \frac{29715.08}{908.08} = 32.72
   \]
   The year corresponding to $ t = 32.72 $ is approximately $ 1980 + 33 = 2013 $.

### Summary:
- The linear model that fits the data is approximately:  
  \[
  I(t) = 908.08t + 8484.92
  \]
- The estimated per capita income in 2015 is **$40,268**.
- The income will reach **$38,200** around the year **2013**.

---

Do you need further details or clarification? Here are five related questions to expand on this:

1. How would the model change if the income growth slowed down in recent years?
2. How do we interpret the slope of 908.08 in the context of income growth?
3. What would happen if the intercept was much lower than 8484.92?
4. Could a non-linear model better explain this data in the long term?
5. How could inflation affect the accuracy of this model over time?

**Tip:** Always verify the linearity of data before applying a linear regression model for prediction!

The user provided a table of personal income per capita for Georgia and asked to estimate a linear model to fit the data, predict future income, and estimate the year in which a specified income will be reached.

Learn how to use linear regression to model personal income data for Georgia. This problem covers key concepts like calculating the slope and intercept, estimating income in future years, and solving for specific income thresholds. Suitable for students in Grades 9-12.

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