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.

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!