Certainly! To determine the final amount of money in an account after depositing [tex]$5,300 at an interest rate of 3% compounded weekly over a period of 9 years, we need to use the compound interest formula:
\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]
Here's a step-by-step breakdown:
1. Identify the variables:
- \( P \) (Principal or initial amount) = $[/tex]5,300
- [tex]\( r \)[/tex] (Annual interest rate) = 3% or 0.03
- [tex]\( n \)[/tex] (Number of compounding periods per year) = 52 (Since it's compounded weekly)
- [tex]\( t \)[/tex] (Time in years) = 9
2. Insert the values into the compound interest formula:
[tex]\[ A = 5300\left(1 + \frac{0.03}{52}\right)^{52 \times 9} \][/tex]
3. Simplify the calculations inside the parentheses:
[tex]\[ \frac{0.03}{52} \approx 0.000576923 \][/tex]
4. Add 1 to this value:
[tex]\[ 1 + 0.000576923 \approx 1.000576923 \][/tex]
5. Raise this result to the power of 52 times 9:
[tex]\[ (1.000576923)^{468} \approx 1.309849268 \][/tex]
6. Multiply this result by the principal amount [tex]$5,300:
\[ 5300 \times 1.309849268 \approx 6942.271079602918 \]
7. Round the final amount to 2 decimal places:
\[ \text{Final Amount} \approx 6942.27 \]
Thus, the final amount in the account after 9 years, with weekly compounding at 3% interest, is approximately $[/tex]6942.27.
