Certainly! Let's determine the future value of an investment of \[tex]$360 at an annual interest rate of 4% compounded quarterly over 18 years.
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate,
- \( n \) is the number of times interest is compounded per year,
- \( t \) is the number of years the money is invested or borrowed for,
- \( A \) is the amount of money accumulated after n years, including interest.
Given:
- \( P = 360 \) (Principal),
- \( r = 0.04 \) (4% annual interest rate),
- \( n = 4 \) (Compounded quarterly),
- \( t = 18 \) (Number of years).
Now, substitute these values into the formula:
\[ A = 360 \left(1 + \frac{0.04}{4}\right)^{4 \times 18} \]
First, calculate the term inside the parentheses:
\[ \frac{0.04}{4} = 0.01 \]
So,
\[ 1 + 0.01 = 1.01 \]
Next, calculate the exponent:
\[ 4 \times 18 = 72 \]
Thus, the expression becomes:
\[ A = 360 \left(1.01\right)^{72} \]
Using the exponentiation:
\[ 1.01^{72} \approx 2.046 \]
So now multiply:
\[ A = 360 \times 2.046 \]
\[ A \approx 736.96 \]
So, after 18 years, the investment will be worth approximately \$[/tex]736.96.
Therefore, the correct answer is \$736.96.
