The problem asks for the accumulated value of an investment of $10,000 for 5 years at an interest rate of 5.5%, where the interest is compounded semiannually. Let's go step-by-step using the compound interest formula:

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

Where:

• $A$ is the accumulated amount (what we are solving for),
• $P = 10,000$ is the principal amount,
• $r = 0.055$ (5.5% interest rate),
• $n = 2$ (since it is compounded semiannually),
• $t = 5$ (the time in years).

Step-by-step Calculation:

1. Substitute the values into the formula: $A = 10,000 \left(1 + \frac{0.055}{2}\right)^{2 \times 5}$

2. Simplify the inside of the parentheses: $A = 10,000 \left(1 + 0.0275\right)^{10}$ $A = 10,000 \left(1.0275\right)^{10}$

3. Raise 1.0275 to the power of 10: $A = 10,000 \times 1.314073$

4. Multiply by the principal: $A = 13,140.73$

Answer:

The accumulated value after 5 years, compounded semiannually, is $13,140.73.

Would you like more details on this calculation or help with other parts of the problem?

Here are 5 related questions:

1. What would the accumulated value be if compounded quarterly?
2. How does continuous compounding change the result?
3. What is the formula for continuous compounding?
4. How does the interest rate impact the accumulated value when compounded more frequently?
5. What is the difference between nominal and effective interest rates?

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