Interest rate. Upstate Bank is offering long-term certificates of deposit with a face value of \( \$ 100,000 \) (future value). Bank customers can buy these CDs today for \( \$ 45,000 \) and will receive the \( \$ 100,000 \) in 20 years. What interest rate is the bank paying on these CDs? What interest rate is the bank paying on these CDs? \( \square \% \) (Round to two decimal plices.)

To find the interest rate that the bank is paying on these CDs, we can use the formula for compound interest: \[ A = P(1 + \frac{r}{100})^n \] where: - \( A \) is the future value of the investment (in this case, $100,000), - \( P \) is the present value of the investment (in this case, $45,000), - \( r \) is the interest rate (which we need to find), - \( n \) is the number of years (in this case, 20 years). Given that the present value of the investment is $45,000 and the future value is $100,000, we can rearrange the formula to solve for the interest rate \( r \): \[ 100,000 = 45,000(1 + \frac{r}{100})^{20} \] Now, we can solve for \( r \) to find the interest rate that the bank is paying on these CDs. Solve the equation by following steps: - step0: Solve for \(r\): \(100000=45000\left(1+\frac{r}{100}\right)^{20}\) - step1: Swap the sides: \(45000\left(1+\frac{r}{100}\right)^{20}=100000\) - step2: Divide both sides: \(\frac{45000\left(1+\frac{r}{100}\right)^{20}}{45000}=\frac{100000}{45000}\) - step3: Divide the numbers: \(\left(1+\frac{r}{100}\right)^{20}=\frac{20}{9}\) - step4: Simplify the expression: \(1+\frac{r}{100}=\pm \sqrt[20]{\frac{20}{9}}\) - step5: Simplify: \(1+\frac{r}{100}=\pm \frac{\sqrt[20]{20\times 19683^{2}}}{3}\) - step6: Separate into possible cases: \(\begin{align}&1+\frac{r}{100}=\frac{\sqrt[20]{20\times 19683^{2}}}{3}\\&1+\frac{r}{100}=-\frac{\sqrt[20]{20\times 19683^{2}}}{3}\end{align}\) - step7: Calculate: \(\begin{align}&r=\frac{100\sqrt[20]{20\times 19683^{2}}-300}{3}\\&r=-\frac{100\sqrt[20]{20\times 19683^{2}}+300}{3}\end{align}\) - step8: Rewrite: \(r_{1}=-\frac{100\sqrt[20]{20\times 19683^{2}}+300}{3},r_{2}=\frac{100\sqrt[20]{20\times 19683^{2}}-300}{3}\) The interest rate that the bank is paying on these CDs is approximately 4.07%.