The problem involves a loan of $14,500 borrowed for 3 years at an interest rate of 2.8% per year compounded semiannually. Let's solve each part systematically.

Key information:

• Principal $P = 14,500$
• Annual interest rate $r = 2.8\% = 0.028$
• Compounding frequency: Semiannually, so $n = 2$ times per year
• Loan period $t = 3$ years

Part a: Amount of the semiannual payment

The formula to calculate the semiannual payment is derived from the formula for an amortized loan payment:

$A = \frac{P \cdot r/n}{1 - (1 + r/n)^{-nt}}$

Where:

• $A$ is the semiannual payment
• $P$ is the principal loan amount
• $r$ is the annual interest rate
• $n$ is the number of compounding periods per year (in this case, 2 for semiannual)
• $t$ is the total loan period in years

Let me calculate this for you.

Part b: Amount of the first payment going towards the principal

To find the portion of the first payment going towards the principal, we need to calculate the interest portion first. The interest for the first period is calculated as:

$\text{Interest} = P \times \frac{r}{n}$

Then, the amount going towards the principal is:

$\text{Principal payment} = A - \text{Interest}$

Part c: Total amount of interest paid

To calculate the total interest paid, we first need to find the total amount paid over the entire loan period:

$\text{Total amount paid} = A \times (n \times t)$

Then subtract the original principal from this value to get the total interest:

$\text{Total interest} = \text{Total amount paid} - P$

I'll compute the exact values now.Here are the results for each part of the question:

a. Amount of the semiannual payment:

The semiannual payment $A$ is approximately $2,536.46.

b. Amount of the first payment going towards the principal:

The amount going towards the principal in the first payment is approximately $2,333.46.

c. Total amount of interest paid:

The total interest paid over the life of the loan is approximately $718.73.

Would you like any further details or explanations?

Here are some related questions you might consider:

1. How would the payment change if the interest rate was compounded monthly?
2. What if the loan period was extended to 5 years? How would that affect the payments and interest?
3. How much interest would be paid if the rate increased to 4%?
4. What portion of each payment goes to principal versus interest over time?
5. How does semiannual compounding differ from continuous compounding in this scenario?

Tip: The more frequently interest is compounded, the more you end up paying in total interest, even if the difference in the rate seems small.