Matilda's friend Hayden found an account that carns \( 3.5 \% \) interest compounded annually. Hayden made an initial deposit of \( \$ 180 \) into his account at the same time Matilda made a deposit of \( \$ 180 \) into her account. After 8 years, how much more moncy will Hayden's initial deposit have carned than will Matilda's initial deposit? (Round your answer to the nearest cent and ignore the dollar sign when writing your response.)

To find out how much more money Hayden's initial deposit will have earned compared to Matilda's after 8 years, we can use the formula for compound interest: \[ A = P(1 + r)^t \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial deposit). - \( r \) is the annual interest rate (decimal). - \( t \) is the number of years the money is invested or borrowed. For both Hayden and Matilda, the initial deposit \( P \) is \( 180 \) dollars, the interest rate \( r \) is \( 3.5\% \) (or \( 0.035 \) in decimal), and the time \( t \) is \( 8 \) years. Calculating the amount for Hayden's account: \[ A_H = 180(1 + 0.035)^8 \] Calculating \( (1 + 0.035)^8 \): \[ (1 + 0.035)^8 \approx 1.316074012 \] Now, substituting this back into the formula: \[ A_H \approx 180 \times 1.316074012 \approx 237.00 \] Now, calculating the amount for Matilda's account, which has the same parameters: \[ A_M = 180(1 + 0.035)^8 \approx 237.00 \] Since both accounts have the same initial deposit, interest rate, and time, the amounts will be the same: \[ A_H = A_M \approx 237.00 \] Now, to find out how much more Hayden's initial deposit has earned than Matilda's, we calculate: \[ A_H - A_M = 237.00 - 237.00 = 0.00 \] Thus, after 8 years, Hayden's initial deposit will have earned the same amount as Matilda's initial deposit. Therefore, the answer is: \[ \boxed{0.00} \]