To solve this, we will use two different formulas for compound interest:

(a) Semiannual Compounding

The formula for compound interest when interest is compounded semiannually is:

$A = P \left( 1 + \frac{r}{n} \right)^{nt}$

Where:

• $A$ is the future value
• $P$ is the principal (initial investment)
• $r$ is the annual interest rate (as a decimal)
• $n$ is the number of compounding periods per year (for semiannual compounding, $n = 2$)
• $t$ is the number of years

Given values:

• $P = 8704.56$
• $r = 5\% = 0.05$
• $n = 2$ (because it is compounded semiannually)
• $t = 8$ years

We can substitute these values into the formula:

$A = 8704.56 \left( 1 + \frac{0.05}{2} \right)^{2 \times 8}$

$A = 8704.56 \left( 1 + 0.025 \right)^{16}$

$A = 8704.56 \times (1.025)^{16}$

Now, we calculate the power term:

$A = 8704.56 \times 1.448$

$A \approx 12,610.14$

So, the future value of the investment when compounded semiannually is approximately $12,610.14.

To find the interest earned, subtract the initial investment from the future value:

$\text{Interest earned} = 12,610.14 - 8704.56 \approx 3915.58$

Thus, the interest earned is approximately $3915.58.

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(b) Continuous Compounding

For continuous compounding, the formula used is:

$A = P e^{rt}$

Where:

• $A$ is the future value
• $P$ is the principal (initial investment)
• $r$ is the annual interest rate (as a decimal)
• $t$ is the number of years
• $e$ is the base of the natural logarithm (approximately 2.718)

Using the given values:

• $P = 8704.56$
• $r = 0.05$
• $t = 8$

Substitute these into the formula:

$A = 8704.56 \times e^{0.05 \times 8}$

$A = 8704.56 \times e^{0.4}$

Using a calculator to find $e^{0.4}$:

$e^{0.4} \approx 1.4918$

$A \approx 8704.56 \times 1.4918 \approx 12,986.91$

Thus, the future value of the investment when compounded continuously is approximately $12,986.91.

To find the interest earned:

$\text{Interest earned} = 12,986.91 - 8704.56 \approx 4282.35$

So, the interest earned with continuous compounding is approximately $4282.35.

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Summary:

• (a) Future Value (semiannually): $12,610.14, Interest earned: $3915.58
• (b) Future Value (continuously): $12,986.91, Interest earned: $4282.35

Would you like to go over the steps in more detail or have any further questions?

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Related Questions:

1. How would the interest earned change if the interest rate were increased to 6%?
2. What happens to the future value if the investment is compounded quarterly instead of semiannually?
3. How does the formula for compound interest change if the interest is compounded monthly?
4. How does the future value compare if the interest is compounded annually instead of continuously?
5. How can you calculate the initial principal amount given the future value, interest rate, and time period?

Tip: When dealing with compound interest, always be mindful of the compounding frequency, as it directly affects how much interest you earn over time. The more frequent the compounding, the higher the future value!