We are given an investment of $8,000 at an annual interest rate of 8%, and we need to calculate the amount in the bank after 11 years for four different compounding scenarios: annually, quarterly, monthly, and continuously.

1. Compounded Annually:

The formula for compound interest is: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ Where:

• $A$ is the final amount.
• $P$ is the initial principal (in this case, 8000).
• $r$ is the annual interest rate (8% = 0.08).
• $n$ is the number of times the interest is compounded per year (for annually, $n = 1$).
• $t$ is the time in years (11 years).

For annually compounded interest: $A = 8000 \left(1 + \frac{0.08}{1}\right)^{1 \times 11}$ $A = 8000 \times (1.08)^{11}$

2. Compounded Quarterly:

For quarterly compounded interest, $n = 4$: $A = 8000 \left(1 + \frac{0.08}{4}\right)^{4 \times 11}$ $A = 8000 \times \left(1 + 0.02\right)^{44}$

3. Compounded Monthly:

For monthly compounded interest, $n = 12$: $A = 8000 \left(1 + \frac{0.08}{12}\right)^{12 \times 11}$ $A = 8000 \times \left(1 + \frac{0.08}{12}\right)^{132}$

4. Compounded Continuously:

For continuously compounded interest, the formula is: $A = P \times e^{rt}$ Where $e$ is Euler's number ($\approx 2.718$), and the rest of the variables remain the same. Thus: $A = 8000 \times e^{0.08 \times 11}$

Let me calculate the final amounts for each case.Here are the calculated amounts for the investment of $8,000 at 8% interest over 11 years under different compounding scenarios:

1. Compounded Annually: $18,653.11
2. Compounded Quarterly: $19,120.43
3. Compounded Monthly: $19,230.95
4. Compounded Continuously: $19,287.20

Would you like more details on any of these calculations or a breakdown of a specific case?

Here are five related questions to expand on this topic:

1. What is the formula for compound interest, and how does it work?
2. How does increasing the frequency of compounding affect the final amount?
3. What happens if the interest rate changes mid-investment?
4. How can you calculate the time it takes for an investment to double at a specific interest rate?
5. How would different initial amounts affect the final balance after compounding?

Tip: The more frequently interest is compounded, the higher the final amount will be.