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 

​$10 comma 00010,000 

deposit in an account with an APR of 

3.53.5​%.

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Part 1

The balance in the account after 

11 

year is approximately 

​$enter your response here.

​(Round to the nearest cent as​ needed.)

To solve this, we use the continuous compounding formula:

\[
A = P \cdot e^{(r \cdot t)}
\]

where:
- $ A $ is the amount in the account after $ t $ years,
- $ P $ is the initial principal (in this case, $10,000),
- $ r $ is the annual interest rate (APR) in decimal form (3.5% or 0.035),
- $ t $ is the time in years,
- $ e $ is the base of the natural logarithm, approximately equal to 2.71828.

### Calculations:

1. **Balance after 1 year ($ t = 1 $)**:
   \[
   A = 10000 \cdot e^{(0.035 \cdot 1)}
   \]

2. **Balance after 5 years ($ t = 5 $)**:
   \[
   A = 10000 \cdot e^{(0.035 \cdot 5)}
   \]

3. **Balance after 20 years ($ t = 20 $)**:
   \[
   A = 10000 \cdot e^{(0.035 \cdot 20)}
   \]

I'll compute these values and then determine the APY (Annual Percentage Yield).

### APY Calculation
The APY can be calculated for continuous compounding as:

\[
APY = e^r - 1
\]

Let's proceed with the calculations.Here are the computed balances and APY for the account:

1. **Balance after 1 year**: $10,356.20
2. **Balance after 5 years**: $11,912.46
3. **Balance after 20 years**: $20,137.53

### APY (Annual Percentage Yield)
The APY for this account is approximately **3.56%**.

Let me know if you’d like further details or explanations. 

### Related Questions
1. How would the balance change with a higher APR?
2. What if compounding was quarterly instead of continuous?
3. How would the APY differ if compounded annually instead of continuously?
4. What would be the balance after 10 years with the same conditions?
5. How is continuous compounding different from simple interest?

### Tip
The APY gives a true annual growth rate, so it’s particularly useful for comparing accounts with different compounding frequencies.

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 $10,000 deposit in an account with an APR of 3.5%. The balance in the account after 1 year is approximately $enter your response here (Round to the nearest cent as needed).

Learn how to calculate the balance in an account with continuous compounding after 1, 5, and 20 years. This example uses a $10,000 deposit at 3.5% APR, covering continuous compounding formulas and APY calculations. Ideal for high school students in Grades 10-12.

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