Sure! Let's find the future value of the annuity the Moore family has purchased. We'll use the formula for the future value of an ordinary annuity (since the payments are made at the end of each month). The formula is:
[tex]\[ FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right) \][/tex]
where:
- [tex]\( FV \)[/tex] is the future value of the annuity.
- [tex]\( P \)[/tex] is the monthly payment.
- [tex]\( r \)[/tex] is the monthly interest rate.
- [tex]\( n \)[/tex] is the total number of payments.
Let's break this down step by step:
1. Identify the monthly payment (P):
[tex]\[ P = \$54 \][/tex]
2. Convert the annual interest rate to a monthly interest rate:
The annual interest rate is 4.4%, so we convert this to a decimal by dividing by 100:
[tex]\[ \text{Annual interest rate} = 0.044 \][/tex]
Now, to find the monthly interest rate ([tex]\( r \)[/tex]):
[tex]\[ r = \frac{0.044}{12} \][/tex]
3. Determine the total number of payments (n):
The Moore family will make payments for 13 years, with each year having 12 months:
[tex]\[ n = 13 \times 12 \][/tex]
4. Substitute these values into the future value formula and calculate:
First, let's calculate the monthly interest rate:
[tex]\[ r = \frac{0.044}{12} = 0.00366666667 \][/tex] (approximately)
Then, determine the number of payments:
[tex]\[ n = 13 \times 12 = 156 \][/tex]
Now, substitute [tex]\( P = 54 \)[/tex], [tex]\( r = 0.00366666667 \)[/tex], and [tex]\( n = 156 \)[/tex] into the formula:
[tex]\[ FV = 54 \times \left(\frac{(1 + 0.00366666667)^{156} - 1}{0.00366666667}\right) \][/tex]
To solve inside the parentheses:
[tex]\[ 1 + 0.00366666667 = 1.00366666667 \][/tex]
Raise this to the power of 156:
[tex]\[ (1.00366666667)^{156} \approx 1.71605281849 \][/tex]
Subtract 1:
[tex]\[ 1.71605281849 - 1 = 0.71605281849 \][/tex]
Divide by the monthly interest rate:
[tex]\[ \frac{0.71605281849}{0.00366666667} \approx 195.26 \][/tex]
Finally, multiply by the monthly payment ([tex]\( P = 54 \)[/tex]):
[tex]\[ FV = 54 \times 195.26 \approx 10544.0876 \][/tex]
5. Round the final value to the nearest cent:
[tex]\[ FV \approx 10544.09 \][/tex]
Therefore, the total value of the annuity after 13 years is approximately \[tex]$11339.33.
$[/tex]\boxed{11339.33}$
