To calculate the balance in the account using the continuous compounding formula, we use the following equation:

$A = P \cdot e^{r \cdot t}$

Where:

• $A$ is the balance after time $t$,
• $P$ is the principal (initial deposit),
• $r$ is the annual percentage rate (APR) as a decimal,
• $t$ is the time in years,
• $e$ is the base of the natural logarithm (approximately 2.71828).

Given values:

• Principal $P = 6000$,
• APR $r = 4.75\% = 0.0475$,
• Time $t = 1$, $5$, and $20$ years.

Part 1: Balance After 1 Year

Using the formula: $A = 6000 \cdot e^{0.0475 \cdot 1} = 6000 \cdot e^{0.0475} \approx 6000 \cdot 1.04867377 \approx 6,291.88$ So, after 1 year, the balance is approximately $6,291.88, which matches the provided result.

Part 2: Balance After 5 Years

Now for $t = 5$ years: $A = 6000 \cdot e^{0.0475 \cdot 5} = 6000 \cdot e^{0.2375} \approx 6000 \cdot 1.268095 \approx 7,608.57$ So, after 5 years, the balance is approximately $7,608.57.

Part 3: Balance After 20 Years

For $t = 20$ years: $A = 6000 \cdot e^{0.0475 \cdot 20} = 6000 \cdot e^{0.95} \approx 6000 \cdot 2.585709 \approx 15,514.25$ So, after 20 years, the balance is approximately $15,514.25.

Part 4: APY Calculation

The Annual Percentage Yield (APY) for continuously compounded interest is given by: $APY = e^r - 1$ Substituting $r = 0.0475$: $APY = e^{0.0475} - 1 \approx 1.04867377 - 1 \approx 0.04867 = 4.867\%$ So, the APY is approximately 4.87%.

Summary of Results:

1. Balance after 1 year: $6,291.88
2. Balance after 5 years: $7,608.57
3. Balance after 20 years: $15,514.25
4. APY: 4.87%

Would you like further clarification or additional details?

5 Related Questions:

1. How would the balances change if the APR was increased to 5%?
2. What is the difference between APR and APY, and why is APY always higher?
3. How do you calculate APY if interest is compounded monthly instead of continuously?
4. What is the effect of doubling the initial deposit on the future balances?
5. How would the balances differ if interest was compounded annually instead of continuously?

Tip: Continuous compounding yields the highest return because interest is calculated and added to the balance at every moment.