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To solve the problem of finding the periodic payment necessary for the deposit to a sinking fund, we can use the formula for the sinking fund payment:

\[
P = \frac{A \cdot i}{(1 + i)^n - 1}
\]

where:
- $ P $ = periodic payment
- $ A $ = amount needed
- $ i $ = periodic interest rate
- $ n $ = total number of payments

### Given:
- $ A = 3,000,000 $
- Annual interest rate $ r = 4\% = 0.04 $
- Time $ t = 22 $ years
- Payments are made semiannually, so the frequency $ n = 2 $ times per year.

### Calculations:

1. **Calculate the periodic interest rate $ i $:**
   \[
   i = \frac{r}{n} = \frac{0.04}{2} = 0.02
   \]

2. **Calculate the total number of payments $ n \cdot t $:**
   \[
   n \cdot t = 2 \times 22 = 44
   \]

3. **Apply these values to the sinking fund formula:**
   \[
   P = \frac{3,000,000 \times 0.02}{(1 + 0.02)^{44} - 1}
   \]

Let's compute this.The periodic payment necessary for the deposit to the sinking fund is approximately **$43,163.82** (rounded to the nearest cent).

Would you like more details or have any questions? Here are some related questions:

1. How would the periodic payment change if the interest rate were different?
2. What is the total amount paid over the 22 years using these periodic payments?
3. How does the compounding frequency affect the periodic payment?
4. What if the payments were made annually instead of semiannually?
5. How does the payment formula change if the sinking fund aims to meet a different future value?

**Tip:** The periodic payment decreases as the interest rate increases because the accumulated interest covers a larger portion of the needed amount.

Find the amount of periodic payment necessary for the deposit to a sinking fund. The amount needed is $3,000,000 with semiannual payments, a 4% annual interest rate, and a 22-year duration.

Learn how to calculate the periodic payments required for a sinking fund deposit of $3,000,000 over 22 years with semiannual payments and a 4% annual interest rate. Step-by-step solution using the sinking fund payment formula.

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