https://file.aiquickdraw.com/mathbot/file/1729642958092eziyzvsk.jpg

To address the questions in the image, let's break it down.

### Part (a): Which curve fits the data best?

By inspecting the three graphs:
- **Figure 1** represents a linear model, $ y = 40x + 400 $.
- **Figure 2** represents an exponential model, $ y = 1.25(1.21)^x $.
- **Figure 3** represents another exponential model, $ y = 583(1.05)^x $.

The data points (blue dots) suggest that the relationship is exponential rather than linear. Between Figures 2 and 3, **Figure 3** shows a much better fit for the higher values of $ y $, closely matching the data points throughout the range. So, **Figure 3** fits the data best.

### Part (b): Predicting the amount of money after 31 years

Now, using the equation from **Figure 3**, which is:
\[
y = 583(1.05)^x
\]
we need to substitute $ x = 31 $ into this equation to predict the amount of money after 31 years.

\[
y = 583(1.05)^{31}
\]
Let me calculate this for you.

\[
y = 583 \times (1.05)^{31} = 583 \times 4.467 = 2604.26
\]

So, after 31 years, the amount of money in the account would be approximately **\$2604.26**.

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Do you want further details or have any questions?

### 5 Related Questions:
1. What does the rate $ 1.05 $ in the exponential model represent in this context?
2. How would the predictions differ using the equation from Figure 2?
3. How can you determine if a relationship is linear or exponential based on data points?
4. How would you calculate the amount for a different number of years, say 50 years?
5. How can you determine if a model fits well just by looking at a graph?

### Tip:
Exponential models grow faster than linear models, especially as the input value increases. Keep an eye on this behavior when interpreting data!

Which curve fits the data best? Use the equation of the best fitting curve to predict the amount of money in the account after 31 years.

Determine which curve best fits a set of data points and use an exponential growth equation to predict the future balance in a bank account after 31 years. Learn key concepts in data analysis, including how to identify and use exponential models. Suitable for high school students.

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