Fincing the cficalive annual interest rate of a ioan or Investment Abdul took out a loan for \( \$ 7200 \) that charges an annual interest rate of \( 9.3 \% \), compounded daily. Assume there are 365 days in each year. Answer each part. If necessary, refer to the list of financial formulas. \[ \begin{array}{l}\text { (a) Find the amount owed after one year, assuming no payments are made. } \\ \text { Do not round any intermediate computations, and round your answer to the nearest cent. } \\ \text { (b) Find the effective annual interest rate, expressed as a percentage. } \\ \text { Do not round any intermediate computations, and round your answer to the nearest hundredth } \\ \text { of a percent. } \\ \square \%\end{array} \]

To find the amount owed after one year, assuming no payments are made, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount owed after \( t \) years, - \( P \) is the principal amount (initial loan amount), - \( r \) is the annual interest rate (in decimal form), - \( n \) is the number of times the interest is compounded per year, - \( t \) is the time in years. Given: - \( P = \$7200 \), - \( r = 9.3\% = 0.093 \), - \( n = 365 \) (compounded daily), - \( t = 1 \) year. Substitute these values into the formula: \[ A = 7200 \left(1 + \frac{0.093}{365}\right)^{365 \cdot 1} \] Now, we can calculate the amount owed after one year. Calculate the value by following steps: - step0: Calculate: \(7200\left(1+\frac{0.093}{365}\right)^{365}\) - step1: Divide the terms: \(7200\left(1+\frac{93}{365000}\right)^{365}\) - step2: Add the numbers: \(7200\left(\frac{365093}{365000}\right)^{365}\) - step3: Simplify: \(7200\times \frac{365093^{365}}{365000^{365}}\) - step4: Rewrite the expression: \(200\times 36\times \frac{365093^{365}}{365000^{365}}\) - step5: Rewrite the expression: \(200\times 36\times \frac{365093^{365}}{200^{365}\times 1825^{365}}\) - step6: Reduce the numbers: \(36\times \frac{365093^{365}}{200^{364}\times 1825^{365}}\) - step7: Rewrite the expression: \(4\times 9\times \frac{365093^{365}}{200^{364}\times 1825^{365}}\) - step8: Rewrite the expression: \(4\times 9\times \frac{365093^{365}}{8^{364}\times 25^{364}\times 1825^{365}}\) - step9: Rewrite the expression: \(2^{2}\times 9\times \frac{365093^{365}}{2^{1092}\times 25^{364}\times 1825^{365}}\) - step10: Reduce the numbers: \(9\times \frac{365093^{365}}{2^{1090}\times 25^{364}\times 1825^{365}}\) - step11: Multiply the fractions: \(\frac{9\times 365093^{365}}{2^{1090}\times 25^{364}\times 1825^{365}}\) The amount owed after one year, assuming no payments are made, is approximately $7901.63. Now, to find the effective annual interest rate, we can use the formula: \[ \text{Effective Annual Interest Rate} = \left(1 + \frac{r}{n}\right)^n - 1 \] Substitute the given values into the formula: \[ \text{Effective Annual Interest Rate} = \left(1 + \frac{0.093}{365}\right)^{365} - 1 \] Now, we can calculate the effective annual interest rate. Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.093}{365}\right)^{365}-1\) - step1: Divide the terms: \(\left(1+\frac{93}{365000}\right)^{365}-1\) - step2: Add the numbers: \(\left(\frac{365093}{365000}\right)^{365}-1\) - step3: Rewrite the expression: \(\frac{365093^{365}}{365000^{365}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{365093^{365}}{365000^{365}}-\frac{365000^{365}}{365000^{365}}\) - step5: Transform the expression: \(\frac{365093^{365}-365000^{365}}{365000^{365}}\) The effective annual interest rate is approximately 9.74%. Therefore, the answers are: (a) The amount owed after one year is approximately $7901.63. (b) The effective annual interest rate is approximately 9.74%.