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.