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Sagot :
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.
### 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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