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Grayson opened a savings account and deposited $400.00. The account earns 4% interest, compounded continuously. If he wants to use the money to buy a new bicycle in 3 years, how much will he be able to spend on the bike?
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.
$


To solve this problem, we will use the formula for continuously compounded interest:

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

Where:
- $ A $ is the amount of money in the account after time $ t $,
- $ P $ is the principal amount ($400),
- $ r $ is the interest rate (4% or 0.04),
- $ t $ is the time (3 years),
- $ e $ is the base of natural logarithms (approximately 2.71828).

Now, substitute the values into the formula:

\[
A = 400 \times e^{(0.04 \times 3)}
\]

Let's calculate this step by step.

Grayson will be able to spend **$451.00** on the bike after 3 years, with continuous compounding at a 4% interest rate.

Do you have any questions or would you like more details on the calculation?

Here are 5 related questions to explore:
1. How would the amount change if the interest rate were 5%?
2. What if Grayson needed the money after 5 years instead of 3 years?
3. How does continuous compounding differ from annual compounding?
4. How much interest did Grayson earn over the 3 years?
5. What is the formula for interest compounded annually instead of continuously?

**Tip:** Continuously compounded interest grows faster than interest compounded annually, quarterly, or monthly due to the frequent calculation of interest.

Grayson opened a savings account and deposited $400.00. The account earns 4% interest, compounded continuously. If he wants to use the money to buy a new bicycle in 3 years, how much will he be able to spend on the bike? 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 Grayson's savings account using the continuously compounded interest formula. Starting with $400 at a 4% interest rate for 3 years, find out how much he will have for his new bicycle. This problem covers exponential growth and continuous compounding, suitable for high school students in Grades 9-12.

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