Let's walk through the problem step-by-step to find the balance after 14 years, given that [tex]$550 is deposited at an annual interest rate of 7.5%, compounded monthly.
### Step 1: Gather all necessary information
- Principal amount (\(P\)): $[/tex]550
- Annual interest rate ([tex]\(r\)[/tex]): 7.5% or 0.075 in decimal form
- Number of times interest is compounded per year ([tex]\(n\)[/tex]): 12 (since it's compounded monthly)
- Number of years the money is invested ([tex]\(t\)[/tex]): 14
### Step 2: Write down the formula for compound interest
The formula for compound interest is:
[tex]\[ F = P \left(1 + \frac{r}{n}\right)^{n \cdot t} \][/tex]
### Step 3: Substitute the given values into the formula
[tex]\[ F = 550 \left(1 + \frac{0.075}{12}\right)^{12 \cdot 14} \][/tex]
### Step 4: Perform the calculations inside the parentheses first
Calculate the monthly interest rate:
[tex]\[ \frac{0.075}{12} = 0.00625 \][/tex]
Add 1 to the monthly interest rate:
[tex]\[ 1 + 0.00625 = 1.00625 \][/tex]
### Step 5: Calculate the exponent
Multiply the number of times interest is compounded per year by the number of years:
[tex]\[ 12 \cdot 14 = 168 \][/tex]
### Step 6: Raise the base to the power of the exponent
Calculate [tex]\( 1.00625^{168} \)[/tex]
### Step 7: Multiply the result by the principal amount
[tex]\[ F = 550 \cdot 1.00625^{168} \][/tex]
### Step 8: Round the result to the nearest cent
[tex]\[ F \approx 1566.58 \][/tex]
### Final Answer
The balance after 14 years, rounded to the nearest cent, is [tex]$\$[/tex]1566.58$.
