Franco puts $400.00 into an account to use for school expenses. The account earns 6% interest, compounded continuously. How much will be in the account after 5 years?
Use the formula A=Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms (≈2.71828), r is the interest rate expressed as a decimal, and t is the time in years.
Round your answer to the nearest cent.

The formula for continuous compounding is:

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

Where:
- $ A $ is the amount after time $ t $,
- $ P $ is the initial principal (starting amount),
- $ r $ is the interest rate as a decimal,
- $ t $ is the time in years, and
- $ e $ is approximately 2.71828.

We are given:
- $ P = 400 $,
- $ r = 0.06 $ (since 6% = 0.06),
- $ t = 5 $ years.

Now, let's calculate the amount $ A $:

\[
A = 400 \times e^{0.06 \times 5}
\]

I'll compute the result.After 5 years, Franco will have **$539.94** in the account.

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

Here are 5 related questions to consider:
1. How would the amount change if the interest rate was 8% instead of 6%?
2. What if the money was invested for 10 years instead of 5?
3. How does continuous compounding compare to annual compounding in this case?
4. If Franco wanted to have $600 in the account after 5 years, how much should he initially invest?
5. What is the formula for continuous compounding, and how is it derived?

**Tip:** Continuous compounding generally results in slightly higher amounts compared to periodic compounding.

Franco puts $400.00 into an account to use for school expenses. The account earns 6% interest, compounded continuously. How much will be in the account after 5 years? Use the formula A=Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms (≈2.71828), r is the interest rate expressed as a decimal, and t is the time in years. Round your answer to the nearest cent.

Learn how to calculate the final amount in an account with $400 invested at a 6% continuous compounding interest rate for 5 years. This problem covers key concepts like exponential growth and continuous interest calculation, using the formula A = Pert. Suitable for students in Grades 9-12.

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