The value of Joseph's collection after $x$ years, given that it appreciates at a rate of 6% per year, can be modeled using the formula for exponential growth:

$V(x) = P \cdot (1 + r)^x$

Where:

• $V(x)$ is the value of the collection after $x$ years.
• $P$ is the initial value of the collection ($460).
• $r$ is the annual appreciation rate (6% = 0.06).
• $x$ is the number of years.

Substituting the given values:

$V(x) = 460 \cdot (1 + 0.06)^x$

Simplify the equation:

$V(x) = 460 \cdot (1.06)^x$

This equation represents the value of the collection after $x$ years.

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Do you have questions or want a graph of this function?

Five Related Questions:

1. What would the collection be worth after 5 years?
2. How long would it take for the collection to double in value?
3. What would happen to the equation if the appreciation rate changed to 4%?
4. How does this equation compare to linear appreciation instead of exponential?
5. Can you calculate the value after a fractional number of years, such as 2.5 years?

Tip:

Exponential growth can result in rapid increases over time, so even small changes in the growth rate significantly affect future values.