$738.56, the interest rate is 5.8%, and the payments are made semiannually for 7 years.

Given:
- Present value (PV) of the loan: \$738.56
- Annual interest rate: 5.8%
- Payments are made semiannually
- Total duration: 7 years

We need to find the semiannual payment amount.

### Step-by-Step Solution:

1. **Convert the annual interest rate to a semiannual interest rate:**

   \[
   \text{Semiannual interest rate} = \frac{5.8\%}{2} = 2.9\% = 0.029
   \]

2. **Calculate the total number of semiannual periods:**

   \[
   \text{Number of periods} = 7 \text{ years} \times 2 = 14 \text{ periods}
   \]

3. **Use the formula for the present value of an ordinary annuity to find the payment amount ($PMT$):**

   \[
   PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}
   \]

   Where:
   - $PV$ is the present value (\$738.56)
   - $r$ is the semiannual interest rate (0.029)
   - $n$ is the total number of periods (14)

   Rearranging the formula to solve for $PMT$:

   \[
   PMT = PV \times \frac{r}{1 - (1 + r)^{-n}}
   \]

4. **Plug in the values:**

   \[
   PMT = 738.56 \times \frac{0.029}{1 - (1 + 0.029)^{-14}}
   \]

5. **Calculate the denominator:**

   \[
   1 - (1 + 0.029)^{-14} = 1 - (1.029)^{-14}
   \]

   \[
   (1.029)^{-14} \approx 0.6577
   \]

   \[
   1 - 0.6577 = 0.3423
   \]

6. **Calculate the fraction:**

   \[
   \frac{0.029}{0.3423} \approx 0.0847
   \]

7. **Finally, calculate the payment amount:**

   \[
   PMT = 738.56 \times 0.0847 \approx 62.58
   \]

So, the semiannual payment amount is approximately $\boxed{62.58}$.

Would you like more details on any part of this solution, or do you have any questions?

### Related Questions:

1. How would the payment amount change if the interest rate were compounded quarterly?
2. What is the total amount paid over the 7-year period?
3. How would you calculate the remaining balance after 4 years?
4. What if the payments were made annually instead of semiannually?
5. How would an increase in the interest rate affect the payment amount?
6. Can you calculate the interest paid over the life of the loan?
7. What is the formula for the future value of an annuity?
8. How would you determine the payment amount if the interest rate was variable?

### Tip:
When dealing with annuity formulas, always ensure that the interest rate and the number of periods are consistent (e.g., both semiannual or both annual).

Learn how to calculate semiannual payments for a loan of $738.56 with a 5.8% interest rate over 7 years. Detailed step-by-step solution using the Present Value of an Annuity formula. Suitable for advanced high school students and anyone interested in finance calculations.

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