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This problem involves performing an exponential regression to model the value of an investment $ V(n) $ over time $ n $ (years). The table presents different values of $ n $ and their corresponding values of the investment $ V(n) $.

Here’s how to approach the solution:

### 1. Determine the Exponential Regression Equation
The general form of the exponential regression equation is:

\[
V(n) = a \cdot b^n
\]

Where:
- $ a $ is the initial value of the investment (when $ n = 0 $),
- $ b $ is the growth factor per year.

Using a regression calculator like Desmos, we can input the given data points and find the values of $ a $ and $ b $. Once you have those values, round $ a $ to two decimal places and $ b $ to three decimal places.

### 2. Percent Increase Per Year
The percent increase per year can be calculated from the growth factor $ b $ using the formula:

\[
\text{Percent Increase} = (b - 1) \times 100
\]

This will give you the average percent growth per year.

### 3. Calculate $ V(20) $
To find $ V(20) $, simply substitute $ n = 20 $ into the exponential regression equation:

\[
V(20) = a \cdot b^{20}
\]

Round the result to two decimal places.

### 4. Interpret the Value After a Specific Number of Years
For example, if you are asked to find the value after 20 years, you would say:

\[
\text{"The value of the investment after 20 years is } V(20) \text{ dollars."}
\]

### 5. Find the Time to Reach $11,500
To find when the investment reaches $ V(n) = 11,500 $, you solve the equation:

\[
11500 = a \cdot b^n
\]

Solve for $ n $ using logarithms:

\[
n = \frac{\log(11500/a)}{\log(b)}
\]

This will give the time it takes for the investment to reach $11,500. Round the answer to two decimal places.

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### Expansion Questions:
1. How does the value of $ b $ affect the shape of the investment growth curve?
2. What happens if the value of $ b $ is less than 1? What would this mean in terms of investment value?
3. Can you identify situations where exponential regression would not be a good model for investment growth?
4. How would the model change if the investment had a compounding interest rate applied more frequently, like monthly?
5. What are some real-world factors that might influence the accuracy of an exponential regression model for predicting investment growth?

### Tip:
Always use a reliable calculator or software to determine the parameters of a regression model, as this ensures precision, especially when rounding is required.

Use Desmos calculator to determine the exponential regression equation that models the set of data. Based on the regression model, calculate the percent increase per year, V(20), and time to reach $11,500.

Solve an exponential regression problem for investment growth. Learn how to model data, calculate percent increase per year, find future values, and determine when an investment will reach $11,500 using logarithms.

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