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This problem involves continuous compounding interest, so we'll use the formula:

\[
A = Pe^{rt}
\]

where:
- $ A $ is the future amount (in this case, $3500),
- $ P $ is the initial principal (what we want to find),
- $ r $ is the interest rate (6% or 0.06),
- $ t $ is the time in years (5 years),
- $ e $ is the base of the natural logarithm, approximately 2.71828.

We want to solve for $ P $, so rearranging the formula:

\[
P = \frac{A}{e^{rt}}
\]

Substituting the values:

\[
P = \frac{3500}{e^{0.06 \times 5}}
\]

Let's calculate this.The amount that should be invested now to reach $3500 in five years at a continuous interest rate of 6% is **$2592.86**.

Would you like more details on the calculation steps or have any questions?

Here are some related questions for further practice:
1. How would the answer change if the interest rate were 5% instead of 6%?
2. What would the initial investment need to be if the goal amount were $5000 instead?
3. How does continuous compounding compare to annual compounding for this problem?
4. How much would the investment grow to if it were kept for 10 years instead of 5 years at 6% interest?
5. How would the calculation change if compounding were quarterly rather than continuous?

**Tip:** Continuous compounding grows investments slightly faster than other compounding frequencies because interest is added at every possible instant.

How much should be invested now at an interest rate of 6% per year, compounded continuously, to have $3500 in five years?

Learn how to calculate the initial investment required to reach $3500 in 5 years, given a continuous interest rate of 6%. This problem demonstrates the application of exponential functions and continuous compounding interest formulas. Suitable for high school students (Grades 9-12).

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