Suppose you owe \( \$ 400 \) on your credit card and you decide to make no new purchases and to make the minimum monthly payment on the account. Assuming that the interest rate on your card is \( 1 \% \) per month the unpaid balance and that the minimum payment is \( 2 \% \) of the total (balance plus interest), your balance after \( t \) months is given by B(t) \( =400 \) ( 0.9898 ). Find your balance at each of the given limes. Complete parts ( e ) below. (a) six months After six months, the balance is \( \$ 37614 \). (Round to the nearest cent as needed.) (b) one year (remember that \( t \) is in months) After one year, the balance is \( \$ \square \) (Round to the nearest cent as needed.)

To find the balance after one year, we need to calculate the balance after 12 months. Given: - Initial balance: $400 - Interest rate: 1% per month - Minimum payment: 2% of the total (balance plus interest) The balance after \( t \) months is given by the formula: \[ B(t) = 400 \times (0.9898)^t \] To find the balance after one year (12 months), we substitute \( t = 12 \) into the formula: \[ B(12) = 400 \times (0.9898)^{12} \] Now, we can calculate the balance after one year. Calculate the value by following steps: - step0: Calculate: \(400\times 0.9898^{12}\) - step1: Convert the expressions: \(400\left(\frac{4949}{5000}\right)^{12}\) - step2: Simplify: \(400\times \frac{4949^{12}}{5000^{12}}\) - step3: Rewrite the expression: \(200\times 2\times \frac{4949^{12}}{5000^{12}}\) - step4: Rewrite the expression: \(200\times 2\times \frac{4949^{12}}{200^{12}\times 25^{12}}\) - step5: Reduce the numbers: \(2\times \frac{4949^{12}}{200^{11}\times 25^{12}}\) - step6: Rewrite the expression: \(2\times \frac{4949^{12}}{8^{11}\times 25^{11}\times 25^{12}}\) - step7: Rewrite the expression: \(2\times \frac{4949^{12}}{2^{33}\times 25^{11}\times 25^{12}}\) - step8: Reduce the numbers: \(1\times \frac{4949^{12}}{2^{32}\times 25^{11}\times 25^{12}}\) - step9: Multiply the fractions: \(\frac{4949^{12}}{2^{32}\times 25^{23}}\) The balance after one year is approximately $353.70.