Certainly! Let's solve this step by step.
Given:
- Principal amount (P) = \[tex]$328,133.32
- Annual interest rate (r) = 6.2%
- Time period (t) = 25 years
Jodi will need this amount to last for 25 years with a monthly withdrawal. First, we need to convert the annual interest rate to a monthly interest rate and then determine the total number of months.
1. Convert annual interest rate to monthly interest rate:
- Annual interest rate = 6.2% per year = 0.062 (as a decimal)
- Monthly interest rate = 0.062 / 12 = 0.0051667
2. Calculate the total number of months:
- Total number of months = 25 years * 12 months/year = 300 months
Next, we use the annuity formula to calculate the monthly income.
The formula for calculating the monthly payment (PMT) is:
\[ PMT = \frac{P \times r}{1 - (1 + r)^{-n}} \]
where:
- PMT is the monthly payment,
- \( P \) is the principal (\$[/tex]328,133.32),
- [tex]\( r \)[/tex] is the monthly interest rate (0.0051667),
- [tex]\( n \)[/tex] is the total number of months (300).
Plugging in the numbers:
[tex]\[ PMT = \frac{328,133.32 \times 0.0051667}{1 - (1 + 0.0051667)^{-300}} \][/tex]
Calculate the denominator first:
[tex]\[ 1 - (1 + 0.0051667)^{-300} \][/tex]
Let's simplify [tex]\((1 + 0.0051667)^{-300}\)[/tex]:
[tex]\[ (1.0051667)^{-300} \][/tex]
Now the equation is:
[tex]\[ PMT = \frac{328,133.32 \times 0.0051667}{1 - 0.207647451} \][/tex]
Simplify the expression:
[tex]\[ PMT = \frac{328,133.32 \times 0.0051667}{0.792352549} \][/tex]
Finally, solve for PMT:
[tex]\[ PMT = \frac{1694.04}{0.792352549} \approx 2154.46 \][/tex]
So, the approximate monthly income Jodi will be able to draw is \[tex]$2,154.
Therefore, the correct answer is:
b. \$[/tex]2,154
