To solve this problem, you need to analyze the increase in the price of gold over time using the information provided. Follow these steps:
1. Initial Price:
- The initial price of the gold ring in the year 2000 is [tex]\( \$590 \)[/tex].
2. Annual Increase:
- The price of gold increases by [tex]\( 35\% \)[/tex] per year.
3. Mathematical Representation of Increase:
- An increase of [tex]\( 35\% \)[/tex] can be represented as multiplying by a factor of [tex]\( 1 + 0.35 = 1.35 \)[/tex] each year.
4. Formula for Price After [tex]\( x \)[/tex] Years:
- To find the price of the gold ring after [tex]\( x \)[/tex] years, we use the compound interest formula, which in this context can be generalized to:
[tex]\[
y = P \times (1 + r)^x
\][/tex]
where:
- [tex]\( y \)[/tex] is the future price of the gold ring.
- [tex]\( P \)[/tex] is the initial price of the gold ring (\$590).
- [tex]\( r \)[/tex] is the annual rate of increase (0.35).
- [tex]\( x \)[/tex] is the number of years after the year 2000.
Given these values, the formula becomes:
[tex]\[
y = 590 \times (1.35)^x
\][/tex]
5. Choice Verification:
- Compare the derived formula to the given choices:
- [tex]\( y = 590(1.35)^x \)[/tex]
- [tex]\( y = 590(0.65)^4 \)[/tex]
- [tex]\( y = 35(0.41)^x \)[/tex]
- [tex]\( y = 35(1.59)^r \)[/tex]
The first equation, [tex]\( y = 590(1.35)^x \)[/tex], matches the formula derived from the given conditions.
6. Conclusion:
- The appropriate equation that represents the price of the gold ring [tex]\( x \)[/tex] years after 2000 is:
[tex]\[
y = 590(1.35)^x
\][/tex]
Thus, the correct choice is:
[tex]\[
\boxed{y=590(1.35)^x}
\][/tex]
