An investment of $5,000 is deposited into an account for which interest is compounded continuously. Complete the table by filling in the amounts to which the investment grows at the indicated times. (Round your answers to the nearest cent.)

*r* = 6%An investment of $5,000 is deposited into an account for which interest is compounded continuously. Complete the table by filling in the amounts to which the investment grows at the indicated times. (Round your answers to the nearest cent.)

*r* = 6%

Time (years)Amount

1

$   **Great job!**

2

$   **Good work!**

3

$   **Good!**

4

$   **Excellent!**

5

$   **Fantastic work!**

6

$

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

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

Where:
- $A$ is the amount of money accumulated after $t$ years, including interest.
- $P$ is the principal amount (initial investment), which is $5,000.
- $r$ is the annual interest rate (6% = 0.06).
- $t$ is the time the money is invested for, in years.
- $e$ is the base of the natural logarithm (approximately 2.71828).

We can now calculate the amounts for each time value.

For $ t = 1, 2, 3, 4, 5, 6 $: 

### $ t = 1 $:
\[
A = 5000 \times e^{0.06 \times 1} = 5000 \times e^{0.06}
\]

### $ t = 2 $:
\[
A = 5000 \times e^{0.06 \times 2} = 5000 \times e^{0.12}
\]

### $ t = 3 $:
\[
A = 5000 \times e^{0.06 \times 3} = 5000 \times e^{0.18}
\]

### $ t = 4 $:
\[
A = 5000 \times e^{0.06 \times 4} = 5000 \times e^{0.24}
\]

### $ t = 5 $:
\[
A = 5000 \times e^{0.06 \times 5} = 5000 \times e^{0.30}
\]

### $ t = 6 $:
\[
A = 5000 \times e^{0.06 \times 6} = 5000 \times e^{0.36}
\]

Let me calculate the exact amounts for each.Here are the amounts to which the $5,000 investment grows at the indicated times, with continuous compounding at a 6% annual interest rate:

- After 1 year: **$5,309.18**
- After 2 years: **$5,637.48**
- After 3 years: **$5,986.09**
- After 4 years: **$6,356.25**
- After 5 years: **$6,749.29**
- After 6 years: **$7,166.65**

Would you like more details or have any questions?

Here are 5 related questions you might find useful:
1. How would the amounts change if the interest rate were 8% instead of 6%?
2. What happens if the investment is compounded annually instead of continuously?
3. How much interest is earned after 6 years compared to the initial principal?
4. How long will it take for the investment to double at a 6% continuous interest rate?
5. How can you calculate the continuous interest for non-integer years, like 2.5 years?

**Tip:** Continuous compounding grows your investment faster compared to other compounding methods, as it calculates interest at every moment.

An investment of $5,000 is deposited into an account for which interest is compounded continuously. Complete the table by filling in the amounts to which the investment grows at the indicated times. (Round your answers to the nearest cent.)

r = 6%

Time (years)

1
$   **Great job!**

2
$   **Good work!**

3
$   **Good!**

4
$   **Excellent!**

5
$   **Fantastic work!**

6
$

Calculate the future value of a $5,000 investment with continuous compounding at a 6% interest rate. Using the formula A = P * e^(rt), find the investment's growth over 1 to 6 years. Detailed step-by-step explanation provided.

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