To solve this problem, we need to use the formula for continuously compounded interest:

$A = P e^{rt}$

Where:

• $A$ is the amount after time $t$,
• $P$ is the initial principal (in this case, $5000),
• $r$ is the interest rate (in this case, 6%, or 0.06 as a decimal),
• $t$ is the time in years, and
• $e$ is the base of the natural logarithm (approximately 2.71828).

Let's go through each year:

For $t = 1$ year:

$A = 5000 \times e^{0.06 \times 1}$ Calculating this:

$A \approx 5000 \times e^{0.06} \approx 5000 \times 1.06183655 \approx 5309.18$

For $t = 2$ years:

$A = 5000 \times e^{0.06 \times 2}$ Calculating this:

$A \approx 5000 \times e^{0.12} \approx 5000 \times 1.12749685 \approx 5637.48$

For $t = 3$ years:

$A = 5000 \times e^{0.06 \times 3}$ Calculating this:

$A \approx 5000 \times e^{0.18} \approx 5000 \times 1.19721736 \approx 5986.09$

For $t = 4$ years:

$A = 5000 \times e^{0.06 \times 4}$ Calculating this:

$A \approx 5000 \times e^{0.24} \approx 5000 \times 1.27124915 \approx 6356.25$

For $t = 5$ years:

$A = 5000 \times e^{0.06 \times 5}$ Calculating this:

$A \approx 5000 \times e^{0.30} \approx 5000 \times 1.34985881 \approx 6749.29$

For $t = 6$ years:

$A = 5000 \times e^{0.06 \times 6}$ Calculating this:

$A \approx 5000 \times e^{0.36} \approx 5000 \times 1.43332941 \approx 7174.77$

Now, the corrected amounts to the nearest cent are:

• $t = 1$ year: $5309.18
• $t = 2$ years: $5637.48
• $t = 3$ years: $5986.09
• $t = 4$ years: $6356.25
• $t = 5$ years: $6749.29
• $t = 6$ years: $7174.77

Feel free to ask for more details or if you have any questions!

Here are 5 related questions:

1. How does the formula for continuously compounded interest differ from other interest formulas?
2. What happens to the investment amount if the interest rate increases?
3. Can the same formula be used for monthly or quarterly compounded interest? Why or why not?
4. How does continuous compounding compare to simple interest over time?
5. What would the investment be after 10 years with continuous compounding?

Tip: Continuously compounded interest always grows faster than any other compounding frequency (like annual or monthly).