This problem involves determining the annual rate of return required for an investment to grow to a certain amount over a specified period. It can be solved using the future value formula for compound interest:

$A = P(1 + r)^t$

Where:

• $A$ is the future value of the investment ($32,548 in this case),
• $P$ is the present value or initial investment ($4,600),
• $r$ is the annual interest rate (what we need to find),
• $t$ is the time in years (14 years in this case).

We can rearrange the formula to solve for $r$:

$r = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1$

Let's calculate the rate of return required.Kirk must earn an annual rate of return of approximately 15% on his investment to accumulate enough money to pay for the $32,548 ring in 14 years.

Would you like further details or explanations on any part of this? Here are some related questions to expand on this topic:

1. How does compound interest differ from simple interest in calculations?
2. What happens if Kirk invests additional money periodically over the 14 years?
3. How do changes in the time frame affect the required interest rate?
4. What investment options might offer a 15% annual return?
5. How would taxes or fees affect Kirk's investment growth?

Tip: When investing, it's crucial to consider both the rate of return and the risk associated with achieving that return. High returns often come with higher risks.