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The problem describes a scenario where you deposit $12,000 in a savings account that offers 7.5% interest compounded continuously for 7 years. We are tasked with finding how much will be in the account after 7 years, rounding to the nearest cent.

To solve this, we use the formula for continuous compounding:

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

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

### Given values:
- $ P = 12,000 $
- $ r = 7.5\% = 0.075 $
- $ t = 7 $ years

Substituting these into the formula:

\[
A = 12,000 \cdot e^{0.075 \times 7}
\]

Let's calculate this step-by-step.

1. Multiply the interest rate and time: 
   \[
   0.075 \times 7 = 0.525
   \]

2. Find $ e^{0.525} $:
   \[
   e^{0.525} \approx 1.69046
   \]

3. Now multiply this by the principal:
   \[
   A = 12,000 \cdot 1.69046 \approx 20,285.52
   \]

Thus, after 7 years, the amount in your account will be **$20,285.52**.

Would you like more details on this calculation or have any questions?

Here are 5 follow-up questions:
1. How would the final amount change if the interest rate were 8%?
2. What would be the amount after 10 years instead of 7?
3. How is continuous compounding different from yearly compounding?
4. What if you deposited $15,000 instead of $12,000?
5. Can you compare continuous compounding with monthly compounding for this case?

**Tip:** Continuous compounding grows money faster than other types of compounding because it assumes the interest is added at every possible moment.

You win $12,000 in the lottery. You decide to deposit the money in a savings account until you graduate from college in 7 years. Your bank can give you an interest rate of 7.5% compounded continuously. How much will you have in your account after 7 years? Round your answer to the nearest cent.

Learn how to calculate the amount in a savings account after 7 years with a 7.5% interest rate compounded continuously. Starting with $12,000, this problem covers continuous compounding interest calculations using the formula A = P * e^(rt). Suitable for Grades 9-12.

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