The formula for continuously compounded interest is given by \( A=P e^{r t} \), where \( A \) is the account balance, \( P \) is the principal, \( r \) is the annual interest rate (expressed as a decimal), and \( t \) is the time in years. If \( \$ 5000 \) is deposited into an account earning an annual interest rate of \( 2.5 \% \), compounded continuously, then what is the account balance after 8 years? Round your answer to the nearest dollar and do not include the \( \$ \) symbol in your answer.

To find the account balance after 8 years with continuously compounded interest, we can use the formula for continuously compounded interest: \[ A = P e^{r t} \] Given: - Principal (\( P \)) = $5000 - Annual interest rate (\( r \)) = 2.5% = 0.025 - Time (\( t \)) = 8 years Substitute the given values into the formula: \[ A = 5000 e^{0.025 \times 8} \] Now, we can calculate the account balance after 8 years using the formula. Calculate the value by following steps: - step0: Calculate: \(5000e^{0.025\times 8}\) - step1: Multiply the numbers: \(5000e^{0.2}\) - step2: Convert the expressions: \(5000e^{\frac{1}{5}}\) - step3: Rewrite the expression: \(5000\sqrt[5]{e}\) The account balance after 8 years with continuously compounded interest is approximately $6107.01. Therefore, the account balance after 8 years is $6107.