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The image provided is related to a linear regression problem that models the value $ V $ of an investment (in dollars) $ n $ years after 1990.

From the problem, it looks like the task involves:

1. Creating a linear regression model from the data in the table.
2. Using that model to predict the value of the investment at different times.
3. Calculating the rate at which the investment's value is decreasing.

### Steps to solve:

1. **Linear regression formula**: The linear regression equation has the general form:
   \[
   V(n) = mn + b
   \]
   where:
   - $ V(n) $ is the value of the investment.
   - $ n $ is the number of years after 1990.
   - $ m $ is the slope (rate of decrease/increase per year).
   - $ b $ is the y-intercept (value at year 0, which is 1990).

2. **Determine the slope (m)**: The slope is the rate at which the value decreases each year. We calculate the slope by taking two points from the table:
   \[
   m = \frac{V(n_2) - V(n_1)}{n_2 - n_1}
   \]

   For example, using the points $ (1, 21692) $ and $ (19, 15340.6) $:
   \[
   m = \frac{15340.6 - 21692}{19 - 1}
   = \frac{-6351.4}{18}
   = -352.86
   \]

3. **Determine the y-intercept (b)**: Now, use the slope and one of the points to find $ b $. Using $ n = 1 $ and $ V(n) = 21692 $:
   \[
   21692 = (-352.86)(1) + b
   \]
   \[
   b = 21692 + 352.86 = 22044.86
   \]

4. **Regression equation**: The equation for the linear regression model is:
   \[
   V(n) = -352.86n + 22044.86
   \]

5. **Value in 1990 (n = 0)**: Using $ n = 0 $ in the regression equation:
   \[
   V(0) = 22044.86
   \]
   This is close to the provided answer of $ 22031.06 $, so rounding explains the slight difference.

6. **Rate of decrease**: The slope of the line $ m = -352.86 $ represents the rate of decrease per year. So, the investment is decreasing by approximately **$352.86 per year**.

Would you like a more detailed breakdown of any step or explanation?

### Related questions:
1. What would the value of the investment be after 10 years using this model?
2. How does rounding affect the precision of the linear regression model?
3. What happens to the investment if the rate of decrease changes in the future?
4. Can the regression equation predict when the investment value will reach zero?
5. How would different methods of regression (like quadratic) affect the outcome?

**Tip**: Always check multiple pairs of points when calculating the slope to ensure consistency and accuracy.

The table below shows the value, V, of an investment (in dollars) n years after 1990. Use the linear regression equation that models the set of data above to answer the questions below.

Explore how to model the value of an investment with linear regression based on a data set showing changes over the years. Learn how to calculate the rate of decrease and the investment value at any point in time.

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