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 on the unpaid balance and that the minimum payment is \( 2 \% \) of the total (balance plus interest), your balance after \( t \) months is given by \( \mathrm{B}(\mathrm{t})=400 \) (0.9898). Find your balance at each of the given times. Complete parts (a) through (e) below. (a) six months After six months, the balance is \( \$ 376.14 \). (Round to the nearest cent as needed.) (b) one year (remember that t is in months) After one year, the balance is \( \$ 353.70 \) (Round to the nearest cent as needed.) (c) four years After four years, the balance is \( \$ \square \) (Round to the nearest cent as needed.)

To find the balance at each of the given times, we need to calculate the balance after six months, one year, and four years. 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: \[ B(t) = 400(0.9898) \] (a) Six months: \[ B(6) = 400(0.9898)^6 \] (b) One year (12 months): \[ B(12) = 400(0.9898)^{12} \] (c) Four years (48 months): \[ B(48) = 400(0.9898)^{48} \] Now, we can calculate the balance at each of the given times. Calculate the value by following steps: - step0: Calculate: \(400\times 0.9898^{6}\) - step1: Convert the expressions: \(400\left(\frac{4949}{5000}\right)^{6}\) - step2: Simplify: \(400\times \frac{4949^{6}}{5000^{6}}\) - step3: Rewrite the expression: \(200\times 2\times \frac{4949^{6}}{5000^{6}}\) - step4: Rewrite the expression: \(200\times 2\times \frac{4949^{6}}{200^{6}\times 25^{6}}\) - step5: Reduce the numbers: \(2\times \frac{4949^{6}}{200^{5}\times 25^{6}}\) - step6: Rewrite the expression: \(2\times \frac{4949^{6}}{8^{5}\times 25^{5}\times 25^{6}}\) - step7: Rewrite the expression: \(2\times \frac{4949^{6}}{2^{15}\times 25^{5}\times 25^{6}}\) - step8: Reduce the numbers: \(1\times \frac{4949^{6}}{2^{14}\times 25^{5}\times 25^{6}}\) - step9: Multiply the fractions: \(\frac{4949^{6}}{16384\times 25^{11}}\) The balance after six months is approximately $376.14. Now, let's 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. Now, let's calculate the balance after four years. Calculate the value by following steps: - step0: Calculate: \(400\times 0.9898^{48}\) - step1: Convert the expressions: \(400\left(\frac{4949}{5000}\right)^{48}\) - step2: Simplify: \(400\times \frac{4949^{48}}{5000^{48}}\) - step3: Rewrite the expression: \(200\times 2\times \frac{4949^{48}}{5000^{48}}\) - step4: Rewrite the expression: \(200\times 2\times \frac{4949^{48}}{200^{48}\times 25^{48}}\) - step5: Reduce the numbers: \(2\times \frac{4949^{48}}{200^{47}\times 25^{48}}\) - step6: Rewrite the expression: \(2\times \frac{4949^{48}}{8^{47}\times 25^{47}\times 25^{48}}\) - step7: Rewrite the expression: \(2\times \frac{4949^{48}}{2^{141}\times 25^{47}\times 25^{48}}\) - step8: Reduce the numbers: \(1\times \frac{4949^{48}}{2^{140}\times 25^{47}\times 25^{48}}\) - step9: Multiply the fractions: \(\frac{4949^{48}}{2^{140}\times 25^{95}}\) The balance after four years is approximately $244.53.