Certainly! Let's walk through the steps to find the balance after 14 years for a principal amount of \[tex]$550, an annual interest rate of 7.5%, compounded monthly.
Given values:
- Principal amount, \( P = \$[/tex]550 \)
- Annual interest rate, [tex]\( r = 7.5\% = 0.075 \)[/tex] (as a decimal)
- Number of times interest is compounded per year, [tex]\( n = 12 \)[/tex] (monthly compounding)
- Number of years, [tex]\( t = 14 \)[/tex]
We use the compound interest formula:
[tex]\[ F = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
1. Substitute the known values into the formula:
[tex]\[ F = 550 \left(1 + \frac{0.075}{12}\right)^{12 \cdot 14} \][/tex]
2. Calculate the monthly interest rate [tex]\( \frac{r}{n} \)[/tex]:
[tex]\[ \frac{0.075}{12} = 0.00625 \][/tex]
3. Add this to 1:
[tex]\[ 1 + 0.00625 = 1.00625 \][/tex]
4. Raise this to the power of [tex]\( nt \)[/tex]:
[tex]\[ 1.00625^{(12 \cdot 14)} = 1.00625^{168} \approx 2.8483295 \][/tex]
5. Multiply this with the principal amount [tex]\( P \)[/tex]:
[tex]\[ 550 \times 2.8483295 \approx 1566.5807174663084 \][/tex]
6. Round this to the nearest cent:
[tex]\[ F \approx 1566.58 \][/tex]
So, the balance after 14 years, rounded to the nearest cent, is:
[tex]\[ \boxed{1566.58} \][/tex]
