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 

​$80008000 

deposit in an account with an APR of 

3.63.6​%

To compute the balance in an account using continuous compounding, we use the formula:

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

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

### Given:
- $ P = 8000 $ dollars,
- APR $ r = 3.6\% = 0.036 $,
- Time $ t = 1, 5, 20 $ years.

#### Step 1: Calculate the balance after 1, 5, and 20 years

We'll calculate each of these step-by-step using the continuous compounding formula.

#### Step 2: Find the APY (Annual Percentage Yield)

The formula to find the APY for continuous compounding is:

\[
APY = e^r - 1
\]

Now let's compute everything.### Results:

1. The balance after **1 year**: $8,293.25
2. The balance after **5 years**: $9,577.74
3. The balance after **20 years**: $16,435.47

### APY (Annual Percentage Yield):
The APY for the account is approximately **3.67%**.

Would you like any further details or explanations?

Here are 5 related questions to expand your understanding:
1. What is the difference between APR and APY?
2. How does continuous compounding compare to annual compounding?
3. How does the time period affect the growth of an investment?
4. Can the formula for continuous compounding be used for any type of account?
5. How is Euler's number $ e $ derived, and why is it important in continuous compounding?

**Tip**: The higher the compounding frequency (e.g., continuous), the greater the amount of interest earned over time!

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. An $8000 deposit in an account with an APR of 3.6%.

Calculate the balance of an $8000 deposit using the continuous compounding formula for 1, 5, and 20 years with an APR of 3.6%. Learn how to compute the Annual Percentage Yield (APY) and understand exponential growth through real-world financial applications. Ideal for high school students and financial enthusiasts.

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