Use the formula for continuous compounding to compute the balance in the account after​ 1, 5, and 20 years.​ Also, find the APY for the account.
A ​$17 comma 000 deposit in an account with an APR of 3.5​%.
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Part 1
The balance in the account after 1 year is approximately ​$
  
enter your response here.
​(Round to the nearest cent as​ needed.)

The formula for continuous compounding is given by:

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

Where:
- $ A $ is the balance in the account after time $ t $.
- $ P $ is the initial deposit (principal).
- $ r $ is the annual interest rate (expressed as a decimal).
- $ t $ is the time in years.
- $ e $ is Euler's number (approximately 2.71828).

In this case:
- $ P = 17000 $
- $ r = 3.5\% = 0.035 $
- $ t = 1, 5, 20 $ (for the respective time periods)

We can now calculate the balance for each case.

Let me calculate the balance after 1, 5, and 20 years.The balances in the account are as follows:

- After 1 year: \$17,605.54
- After 5 years: \$20,251.19
- After 20 years: \$34,233.80

Next, we will calculate the APY (Annual Percentage Yield). The APY for continuous compounding is calculated as:

\[
APY = e^r - 1
\]

Let me compute the APY for this account.The APY (Annual Percentage Yield) for the account is approximately 3.56%.

Would you like more details or have any questions about these calculations?

Here are 5 related questions to explore:
1. How does the continuous compounding formula differ from other compounding methods?
2. How would the balances change with a different interest rate?
3. How is APY different from APR (Annual Percentage Rate)?
4. What happens if we change the compounding frequency to daily or monthly?
5. How does continuous compounding benefit long-term investments?

**Tip**: Continuous compounding results in the highest possible return compared to other compounding methods.

Use the formula for continuous compounding to compute the balance in the account after 1, 5, and 20 years. Also, find the APY for the account. A $17,000 deposit in an account with an APR of 3.5%.

Learn how to calculate the balance of a $17,000 deposit in an account with a 3.5% APR using the continuous compounding formula. Find the balance after 1, 5, and 20 years, and discover how to compute the APY. This problem is ideal for high school students and covers key financial math concepts like exponential growth and interest rates.

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