To solve this problem, we'll use the compound interest formula. The formula to calculate the amount of money accumulated after a certain time when interest is compounded periodically is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the final amount of money in the account.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money), which is [tex]$6,000 in this case.
- \( r \) is the annual interest rate (decimal), which is 3.5% or 0.035 as a decimal.
- \( n \) is the number of times the interest is compounded per year, which is weekly, so 52 times.
- \( t \) is the number of years the money is invested, which is 6 years.
Let's plug the values into the formula:
1. \( P = 6000 \)
2. \( r = 0.035 \)
3. \( n = 52 \)
4. \( t = 6 \)
Substituting these values into the formula:
\[ A = 6000 \left(1 + \frac{0.035}{52}\right)^{52 \times 6} \]
\[ A = 6000 \left(1 + \frac{0.035}{52}\right)^{312} \]
Now, let's calculate the part inside the parentheses first:
\[ 1 + \frac{0.035}{52} = 1 + 0.000673076923 = 1.000673076923 \]
Next, raise this to the power of \( 52 \times 6 = 312 \):
\[ \left(1.000673076923\right)^{312} \approx 1.233591512 \]
Finally, multiply this result by the principal amount:
\[ A = 6000 \times 1.233591512 \approx 7401.55 \]
Therefore, the final amount of money in the account after 6 years, rounded to 2 decimal places, is $[/tex]7,401.55.
