Let's solve each of these cases step by step.

The formula for compound interest is:

1. Compound Interest (Discrete Compounding)

$A = P \left(1 + \frac{r}{n}\right)^{n \cdot t}$ Where:

• $A$ = Amount after $t$ years
• $P$ = Principal amount = $8000
• $r$ = Annual interest rate = 10% = 0.10
• $n$ = Number of compounding periods per year
• $t$ = Time in years = 11

1.1. Compounded Annually ($n = 1$):

$A = 8000 \left(1 + \frac{0.10}{1}\right)^{1 \cdot 11}$

1.2. Compounded Quarterly ($n = 4$):

$A = 8000 \left(1 + \frac{0.10}{4}\right)^{4 \cdot 11}$

1.3. Compounded Monthly ($n = 12$):

$A = 8000 \left(1 + \frac{0.10}{12}\right)^{12 \cdot 11}$

2. Compound Interest (Continuous Compounding)

For continuous compounding, the formula is: $A = P \cdot e^{r \cdot t}$ Where:

• $e$ is the mathematical constant (approximately 2.718)

2.1. Continuous Compounding:

$A = 8000 \cdot e^{0.10 \cdot 11}$

Let me calculate each value for you. Given values P = 8000 # Principal r = 0.10 # Annual interest rate (10%) t = 11 # Time in years

Compounding frequencies

n_annual = 1 # Annual compounding n_quarterly = 4 # Quarterly compounding n_monthly = 12 # Monthly compounding

Continuous compounding uses the exponential function

import math

Calculations

A_annual = P * (1 + r / n_annual) ** (n_annual * t) # Annually A_quarterly = P * (1 + r / n_quarterly) ** (n_quarterly * t) # Quarterly A_monthly = P * (1 + r / n_monthly) ** (n_monthly * t) # Monthly A_continuous = P * math.exp(r * t) # Continuous

A_annual, A_quarterly, A_monthly, A_continuousHere are the results for each scenario:

1. Compounded Annually: $22,824.93
2. Compounded Quarterly: $23,710.46
3. Compounded Monthly: $23,924.03
4. Compounded Continuously: $24,033.33

Would you like further details on how these were calculated or need additional assistance?

Related Questions:

1. What would the amount be if the principal was $10,000 instead of $8,000?
2. How does increasing the interest rate to 12% affect the results?
3. What happens if the investment period is reduced to 5 years?
4. How does the number of compounding periods per year influence the final amount?
5. How does continuous compounding differ mathematically from discrete compounding?

Tip:

The more frequently interest is compounded, the higher the total amount due to the effect of earning "interest on interest." Continuous compounding provides the theoretical maximum return.