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Find the final amount in the following retirement account, in which the rate of return on the account and the regular contribution change over time.

[tex]$611 per month invested at 5%, compounded monthly, for 4 years; then $[/tex]737 per month invested at 6%, compounded monthly, for 4 years.

What is the amount in the account after 8 years?

(Do not round until the final answer. Then round to the nearest dollar as needed.)

Sagot :

GinnyAnsweridn
To find the final amount in the retirement account, we need to break the problem down into two phases, because the contribution amount and the rate of return change after 4 years. We'll use the concept of compound interest, where interest is compounded monthly.

### Phase 1: First 4 Years
- Monthly contribution: [tex]$611 - Annual rate of return: 5% (which is equivalent to 0.05 as a decimal) - Number of years: 4 - Compounding frequency: Monthly (12 times a year) #### Step 1: Calculate the monthly interest rate \[ \text{Monthly interest rate} = \frac{5\%}{12} = \frac{0.05}{12} \] #### Step 2: Calculate the number of compounding periods \[ \text{Number of periods} = 4 \text{ years} \times 12 \text{ months/year} = 48 \text{ periods} \] #### Step 3: Calculate the future value of the annuity for the first 4 years We use the future value of an annuity formula: \[ FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right) \] where \( P \) is the monthly contribution, \( r \) is the monthly interest rate, and \( n \) is the number of compounding periods. \[ FV_1 = 611 \times \left(\frac{(1 + \frac{0.05}{12})^{48} - 1}{\frac{0.05}{12}}\right) \approx 32527.06192284811 \] ### Phase 2: Next 4 Years In this phase, the new amount accumulated from the first phase acts as the principal sum for the next phase. - Monthly contribution: $[/tex]737
- Annual rate of return: 6% (which is equivalent to 0.06 as a decimal)
- Number of years: 4
- Compounding frequency: Monthly (12 times a year)

#### Step 4: Calculate the new monthly interest rate for the second phase
[tex]\[ \text{Monthly interest rate} = \frac{6\%}{12} = \frac{0.06}{12} \][/tex]

#### Step 5: Calculate the number of compounding periods for the second phase
[tex]\[ \text{Number of periods} = 4 \text{ years} \times 12 \text{ months/year} = 48 \text{ periods} \][/tex]

### Step 6: Calculate the future value of the annuity for the next 4 years, taking into account the first phase amount as the principal
We first need to take the final amount from the first 4 years as the principal (initial amount):

[tex]\[ FV_2 = (32527.06192284811 + 737 \times \left(\frac{(1 + \frac{0.06}{12})^{48} - 1}{\frac{0.06}{12}}\right) \approx 81394.73247244436 \][/tex]

### Final Step: Round the final result to the nearest dollar
[tex]\[ \text{Final Amount} = 81395 \][/tex]

Thus, the amount in the account after 8 years, rounded to the nearest dollar, is $81,395.
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