To solve this problem, we need to deal with two parts: finding how much should be deposited for the first 15 years and how much interest is earned over the entire 35-year process (Part A), and then determining how much can be withdrawn monthly if the deposit amount is fixed (Part B).

Let's break it down:

Part (A)

We are dealing with two annuities:

1. Accumulation phase: An ordinary annuity where monthly deposits are made over the first 15 years (180 months).
2. Withdrawal phase: An ordinary annuity where withdrawals are made over the next 20 years (240 months), reducing the balance to zero.

The interest rate is 6.48% compounded monthly, so the monthly interest rate $i$ is:

$i = \frac{6.48\%}{12} = 0.54\% = 0.0054$

1. Accumulation Phase (Deposits for 15 years)

The formula for the future value $F$ of an ordinary annuity is:

$F = P \times \frac{(1 + i)^n - 1}{i}$

Where:

• $P$ is the monthly deposit,
• $i = 0.0054$ is the monthly interest rate,
• $n = 180$ is the number of deposits (15 years $\times$ 12 months).

At the end of 15 years, the person will start withdrawing $1,500 per month for 20 years. The future value $F$ of the annuity must be equal to the present value needed for these withdrawals.

2. Withdrawal Phase (Withdrawals for 20 years)

The formula for the present value $PV$ of an ordinary annuity (used to calculate the amount that must be present to make equal withdrawals) is:

$PV = W \times \frac{1 - (1 + i)^{-n}}{i}$

Where:

• $W = 1500$ is the monthly withdrawal amount,
• $i = 0.0054$ is the monthly interest rate,
• $n = 240$ is the number of withdrawals (20 years $\times$ 12 months).

The present value $PV$ is the amount that must be accumulated at the end of the 15-year period.

Step-by-Step Process

1. Calculate the present value needed at the start of the withdrawal phase.
2. Use that value to determine the monthly deposit required during the first 15 years.
3. Calculate the total interest earned over the 35 years.

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Part (B)

In this part, we are given that the monthly deposit is fixed at $1,000 for the first 15 years. We need to find out how much can be withdrawn monthly over the next 20 years.

1. Calculate the future value of the annuity during the deposit phase with a fixed monthly deposit of $1,000.
2. Use that future value to determine the monthly withdrawal amount for the next 20 years.

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Let's perform the calculations step by step.### Part (A) Results:

1. Monthly deposit for the first 15 years: The person should deposit approximately $664.99 each month for 15 years.
2. Total interest earned over the 35-year process: The total interest earned during this period is approximately $240,302.67.

Part (B) Result:

If the person makes monthly deposits of $1,000 for the first 15 years, they can withdraw approximately $2,255.69 per month for the next 20 years.

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Do you want more detailed steps or explanations for any part of this? Here are some related questions to expand on this topic:

1. How is the total interest calculated in such long-term annuities?
2. What if the interest rate were compounded quarterly instead of monthly?
3. How does increasing the deposit duration (e.g., to 20 years) affect the withdrawal phase?
4. What happens if the interest rate changes during the deposit or withdrawal phase?
5. How can we calculate the remaining balance after a certain number of withdrawals?

Tip: Small differences in interest rates and deposit amounts can significantly impact long-term savings or withdrawal plans. Always review financial projections regularly!