To appropriately interpret the model, we need to look at the characteristics of an exponential growth function, typically represented in the form \( V(t) = V_0 \times e^{rt} \) or \( V(t) = V_0 \times (1 + r)^t \), where:
- \( V(t) \) is the value at time \( t \),
- \( V_0 \) is the initial value,
- \( r \) is the growth rate (expressed as a decimal),
- \( t \) is the number of years.
If the function indicates that the painting's value increases by a certain percentage (like 15%), it would mean that each year the value is multiplied by 1.15 (which translates to a 15% increase).
Evaluating the answer choices:
A. Each year the painting increases its value by 12,000 percent.
- This is incorrect as 12,000% increase is unrealistically high for an annual increase.
B. The painting was sold at the auction 12,000 years ago.
- This is incorrect as it provides a specific past time that is unrelated to the model.
C. Each year the painting increases its value by 15 percent.
- This statement would be correct given that a common interpretation of exponential growth with \( r = 0.15 \) (15%) means that each year the value is multiplied by 1.15.
D. The painting was sold at the auction 1.15 years ago.
- This statement incorrectly interprets a numerical value in a context that does not correlate to the time elapsed since the auction.
Given this analysis, if the model indeed indicates a growth rate consistent with 15%, then the correct interpretation would be:
C. Each year the painting increases its value by 15 percent.
1 answer