Solution for Part (a): Identifying the Best-Fitting Curve

From the three figures:

• Figure 1 (exponential model $y = 10(1.17)^x$) underestimates the values of the data points significantly as $x$ increases.
• Figure 3 (linear model $y = 88.1x - 665$) shows a linear relationship, but the data clearly follows a curve (non-linear growth), so this doesn't fit the data well.
• Figure 2 (exponential model $y = 51(1.13)^x$) closely matches the trend of the data points with a reasonable fit.

Best Fit: Figure 2

Solution for Part (b): Predicting the Value After 11 Years

We use the equation from Figure 2: $y = 51(1.13)^x$ Substitute $x = 11$:

$y = 51(1.13)^{11}$

1. Calculate $1.13^{11}$ using a calculator: $1.13^{11} \approx 3.698$

2. Multiply by 51: $y = 51 \times 3.698 \approx 188.6$

Final Answer:

The value of the item 11 years after purchase is approximately 188.60 dollars.

Clarifications or Additional Questions?

Would you like a step-by-step breakdown of the exponential calculation?

Related Questions:

1. What is the general form of an exponential growth equation?
2. How can you determine which model (exponential or linear) fits a dataset best?
3. Why does exponential growth apply to certain real-world data more than linear models?
4. How would the curve change if the base of the exponent was smaller (e.g., $1.05$ instead of $1.13$)?
5. Can you explain how doubling time relates to exponential growth?

Tip:

Exponential growth occurs when a quantity increases by a fixed percentage over equal intervals of time. To quickly estimate growth, focus on the base of the exponent (like $1.13$ here) as it controls the rate of growth.