9. Kaylee used her graduation money to set up a savings account that earns \( 3.4 \% \) interest compounded weekly. If the original amount deposited was \( \$ 500 \), how much interest will she have earned after 10 years? \( P=50.50 .(1.00 \square 653846)^{52} \)

To find out how much interest Kaylee will have earned after 10 years, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial deposit or investment). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed for. Here, \( P = 500 \), \( r = 0.034 \), \( n = 52 \) (weekly compounding), and \( t = 10 \). Now we plug the values into the formula: \[ A = 500 \left(1 + \frac{0.034}{52}\right)^{52 \times 10} \] Calculating inside the parentheses first: \[ A = 500 \left(1 + 0.000653846\right)^{520} \] Now calculate \( A \): \[ A \approx 500 \left(1.000653846\right)^{520} \approx 500 \times 1.392364 \approx 696.18 \] The interest earned is: \[ \text{Interest} = A - P = 696.18 - 500 = 196.18 \] Kaylee will have earned approximately \$196.18 in interest after 10 years. To make the math even more engaging, consider this: If Kaylee's savings account were a garden, her graduation money would be the seeds she planted. With diligent care (like letting the interest work for her), her small investment will flourish into a substantial amount over the years! Additionally, the idea of compounded interest is beautifully powerful! What might start as a modest sum can grow substantially over time, showcasing why starting early with savings can make a significant difference in financial health. Kaylee's decision? A smart one for her future financial growth!