To determine the future value of an investment when interest is compounded, we use the compound interest formula:
[tex]\[ A = P\left(1+\frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial sum of money).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years the money is invested for.
Let’s break down each given value and substitute it into the formula:
- The principal amount [tex]\( P \)[/tex] is \[tex]$180.
- The annual interest rate \( r \) is 7%, which is 0.07 in decimal form.
- The number of times interest is compounded per year \( n \) is 52 (since it is compounded weekly).
- The number of years \( t \) is 12.
Substituting these values into the formula, we get:
\[ A = 180\left(1+\frac{0.07}{52}\right)^{52 \times 12} \]
Simplify the fraction inside the parentheses:
\[ \frac{0.07}{52} \approx 0.001346153846 \]
Thus, the expression becomes:
\[ A = 180 \left(1 + 0.001346153846\right)^{624} \]
Add the 1 inside the parentheses:
\[ A = 180 \left(1.001346153846\right)^{624} \]
Now, we need to calculate the value of \( \left(1.001346153846\right)^{624} \). Carrying out this computation (either manually or using a calculator), we get approximately:
\[ 1.001346153846^{624} \approx 2.315058882 \]
Finally, multiplying this result by the principal amount:
\[ A = 180 \times 2.315058882 \approx 416.71059878726584 \]
Therefore, the future value of the investment after 12 years, when compounded weekly at an annual interest rate of 7%, is approximately \$[/tex]416.71.
So, the correct answer from the given options is:
[tex]\[ \boxed{\$ 416.71} \][/tex]
