Find the accumulated value of an investment of $15000 for 5 years at an interest rate of 1.15% if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d compounded continuously. What is the accumulated value if the money is compounded semiannually, rounding to the nearest cent if needed?

To solve this problem, we'll use the compound interest formula for parts (a) through (c) and the formula for continuous compounding for part (d).

### 1. Compound Interest Formula (for a, b, c)
The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $ A $ = Accumulated value (amount after interest)
- $ P $ = Principal amount (initial investment) = $15,000
- $ r $ = Annual interest rate (as a decimal) = 0.0115
- $ n $ = Number of times the interest is compounded per year
- $ t $ = Time the money is invested for in years = 5 years

### 2. Continuous Compounding Formula (for d)
The formula for continuous compounding is:
\[
A = P e^{rt}
\]
Where:
- $ e $ is the base of the natural logarithm (approximately 2.71828)
  
Now, let's calculate each part.

### Part (a) Compounded Semiannually
- $ n = 2 $ (because semiannually means twice a year)

\[
A = 15000 \left(1 + \frac{0.0115}{2}\right)^{2 \times 5}
\]

### Part (b) Compounded Quarterly
- $ n = 4 $ (because quarterly means four times a year)

\[
A = 15000 \left(1 + \frac{0.0115}{4}\right)^{4 \times 5}
\]

### Part (c) Compounded Monthly
- $ n = 12 $ (because monthly means twelve times a year)

\[
A = 15000 \left(1 + \frac{0.0115}{12}\right)^{12 \times 5}
\]

### Part (d) Compounded Continuously
- For continuous compounding:

\[
A = 15000 \times e^{0.0115 \times 5}
\]

I'll calculate these values.Here are the accumulated values for the investment after 5 years:

- **(a) Compounded Semiannually:** $15,885.16
- **(b) Compounded Quarterly:** $15,886.47
- **(c) Compounded Monthly:** $15,887.34
- **(d) Compounded Continuously:** $15,887.78

### Answer to the Semiannual Compounding (Rounding to the nearest cent):
The accumulated value when compounded semiannually is **$15,885.16**.

Would you like more details or have any questions? 

Here are some related questions you might find interesting:
1. How does the frequency of compounding affect the accumulated value?
2. What would be the accumulated value after 10 years under the same conditions?
3. How would the accumulated value change if the interest rate were 2% instead of 1.15%?
4. What if the investment amount was $20,000 instead of $15,000, how would the values differ?
5. How do different interest rates impact the difference between semiannual and continuous compounding?

**Tip:** Compounding more frequently results in a slightly higher accumulated value due to interest being calculated on interest more often.

Calculate the accumulated value of a $15,000 investment over 5 years at a 1.15% interest rate compounded semiannually, quarterly, monthly, and continuously. Detailed solutions provided with rounded values. Ideal for understanding compound interest calculations in financial mathematics.

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