Answer :
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]
[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]
