Interest rate (with changing years). Keiko is looking at the following investment choices and wants to know what annual rate of return each choice produces. a. Invest \( \$ 450.00 \) and receive \( \$ 746.23 \) in 11 years. b. Invest \( \$ 3,600.00 \) and receive \( \$ 10,465.82 \) in 17 years. c. Invest \( \$ 32,914.85 \) and receive \( \$ 120,000.00 \) in 20 years. d. Invest \( \$ 33,619.78 \) and receive \( \$ 1,300,000.00 \) in 40 years. b. What annual rate of return will Keiko earn if she invests \( \$ 3,600.00 \) today and receives \( \$ 10,465.82 \) in 17 years? \% (Round to two decimal places.)

To find the annual rate of return for each investment choice, 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 amount of money). - \( r \) is the annual interest rate (decimal). - \( t \) is the time the money is invested for in years. We can rearrange this formula to solve for \( r \): \[ r = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1 \] Now, let's calculate the annual rate of return for each investment choice step by step. ### a. Investment of \( \$ 450.00 \) to receive \( \$ 746.23 \) in 11 years 1. \( P = 450 \) 2. \( A = 746.23 \) 3. \( t = 11 \) Now, we can calculate \( r \): \[ r = \left( \frac{746.23}{450} \right)^{\frac{1}{11}} - 1 \] ### b. Investment of \( \$ 3,600.00 \) to receive \( \$ 10,465.82 \) in 17 years 1. \( P = 3600 \) 2. \( A = 10465.82 \) 3. \( t = 17 \) Now, we can calculate \( r \): \[ r = \left( \frac{10465.82}{3600} \right)^{\frac{1}{17}} - 1 \] ### c. Investment of \( \$ 32,914.85 \) to receive \( \$ 120,000.00 \) in 20 years 1. \( P = 32914.85 \) 2. \( A = 120000 \) 3. \( t = 20 \) Now, we can calculate \( r \): \[ r = \left( \frac{120000}{32914.85} \right)^{\frac{1}{20}} - 1 \] ### d. Investment of \( \$ 33,619.78 \) to receive \( \$ 1,300,000.00 \) in 40 years 1. \( P = 33619.78 \) 2. \( A = 1300000 \) 3. \( t = 40 \) Now, we can calculate \( r \): \[ r = \left( \frac{1300000}{33619.78} \right)^{\frac{1}{40}} - 1 \] Now, I will calculate the values of \( r \) for all four investments. Calculate the value by following steps: - step0: Calculate: \(\left(\frac{120000}{32914.85}\right)^{\frac{1}{20}}-1\) - step1: Divide the numbers: \(\left(\frac{2400000}{658297}\right)^{\frac{1}{20}}-1\) - step2: Rewrite the expression: \(\frac{2400000^{\frac{1}{20}}}{658297^{\frac{1}{20}}}-1\) - step3: Calculate: \(\frac{2400000^{\frac{1}{20}}\times 658297^{\frac{19}{20}}}{658297}-1\) - step4: Reduce fractions to a common denominator: \(\frac{2400000^{\frac{1}{20}}\times 658297^{\frac{19}{20}}}{658297}-\frac{658297}{658297}\) - step5: Transform the expression: \(\frac{2400000^{\frac{1}{20}}\times 658297^{\frac{19}{20}}-658297}{658297}\) - step6: Simplify: \(\frac{\sqrt[20]{2400000}\times \sqrt[20]{658297^{19}}-658297}{658297}\) - step7: Expand the expression: \(\frac{\sqrt[20]{2400000\times 658297^{19}}-658297}{658297}\) Calculate or simplify the expression \( (746.23/450)^(1/11) - 1 \). Calculate the value by following steps: - step0: Calculate: \(\left(\frac{746.23}{450}\right)^{\frac{1}{11}}-1\) - step1: Divide the terms: \(\left(\frac{74623}{45000}\right)^{\frac{1}{11}}-1\) - step2: Rewrite the expression: \(\frac{74623^{\frac{1}{11}}}{45000^{\frac{1}{11}}}-1\) - step3: Calculate: \(\frac{74623^{\frac{1}{11}}\times 45000^{\frac{10}{11}}}{45000}-1\) - step4: Reduce fractions to a common denominator: \(\frac{74623^{\frac{1}{11}}\times 45000^{\frac{10}{11}}}{45000}-\frac{45000}{45000}\) - step5: Transform the expression: \(\frac{74623^{\frac{1}{11}}\times 45000^{\frac{10}{11}}-45000}{45000}\) - step6: Simplify: \(\frac{\sqrt[11]{74623}\times \sqrt[11]{45000^{10}}-45000}{45000}\) - step7: Expand the expression: \(\frac{\sqrt[11]{74623\times 45000^{10}}-45000}{45000}\) Calculate or simplify the expression \( (1300000/33619.78)^(1/40) - 1 \). Calculate the value by following steps: - step0: Calculate: \(\left(\frac{1300000}{33619.78}\right)^{\frac{1}{40}}-1\) - step1: Divide the numbers: \(\left(\frac{65000000}{1680989}\right)^{\frac{1}{40}}-1\) - step2: Rewrite the expression: \(\frac{65000000^{\frac{1}{40}}}{1680989^{\frac{1}{40}}}-1\) - step3: Calculate: \(\frac{65000000^{\frac{1}{40}}\times 1680989^{\frac{39}{40}}}{1680989}-1\) - step4: Reduce fractions to a common denominator: \(\frac{65000000^{\frac{1}{40}}\times 1680989^{\frac{39}{40}}}{1680989}-\frac{1680989}{1680989}\) - step5: Transform the expression: \(\frac{65000000^{\frac{1}{40}}\times 1680989^{\frac{39}{40}}-1680989}{1680989}\) - step6: Simplify: \(\frac{\sqrt[40]{65000000}\times \sqrt[40]{1680989^{39}}-1680989}{1680989}\) - step7: Expand the expression: \(\frac{\sqrt[40]{65000000\times 1680989^{39}}-1680989}{1680989}\) Calculate or simplify the expression \( (10465.82/3600)^(1/17) - 1 \). Calculate the value by following steps: - step0: Calculate: \(\left(\frac{10465.82}{3600}\right)^{\frac{1}{17}}-1\) - step1: Divide the terms: \(\left(\frac{523291}{180000}\right)^{\frac{1}{17}}-1\) - step2: Rewrite the expression: \(\frac{523291^{\frac{1}{17}}}{180000^{\frac{1}{17}}}-1\) - step3: Calculate: \(\frac{523291^{\frac{1}{17}}\times 180000^{\frac{16}{17}}}{180000}-1\) - step4: Reduce fractions to a common denominator: \(\frac{523291^{\frac{1}{17}}\times 180000^{\frac{16}{17}}}{180000}-\frac{180000}{180000}\) - step5: Transform the expression: \(\frac{523291^{\frac{1}{17}}\times 180000^{\frac{16}{17}}-180000}{180000}\) - step6: Simplify: \(\frac{\sqrt[17]{523291}\times \sqrt[17]{180000^{16}}-180000}{180000}\) - step7: Expand the expression: \(\frac{\sqrt[17]{523291\times 180000^{16}}-180000}{180000}\) Round the number \( 0.064788 \) into 2 decimal places. Round the number by following steps: - step0: Round \(0.064788\) to \(2\) decimal places using standard rounding method: \(0.064788\) - step1: Round \(0.064788\) to \(2\) decimal places using standard rounding method\(:\) \(0.06\) Round the number \( 0.047054 \) into 2 decimal places. Round the number by following steps: - step0: Round \(0.047054\) to \(2\) decimal places using standard rounding method: \(0.047054\) - step1: Round \(0.047054\) to \(2\) decimal places using standard rounding method\(:\) \(0.05\) Round the number \( 0.09568 \) into 2 decimal places. Round the number by following steps: - step0: Round \(0.09568\) to \(2\) decimal places using standard rounding method: \(0.09568\) - step1: Round \(0.09568\) to \(2\) decimal places using standard rounding method\(:\) \(0.10\) Round the number \( 0.066816 \) into 2 decimal places. Round the number by following steps: - step0: Round \(0.066816\) to \(2\) decimal places using standard rounding method: \(0.066816\) - step1: Round \(0.066816\) to \(2\) decimal places using standard rounding method\(:\) \(0.07\) Here are the annual rates of return for each investment choice: ### a. Investment of \( \$ 450.00 \) to receive \( \$ 746.23 \) in 11 years - Annual Rate of Return: **5.00%** ### b. Investment of \( \$ 3,600.00 \) to receive \( \$ 10,465.82 \) in 17 years - Annual Rate of Return: **6.00%** ### c. Investment of \( \$ 32,914.85 \) to receive \( \$ 120,000.00 \) in 20 years - Annual Rate of Return: **7.00%** ### d. Investment of \( \$ 33,619.78 \) to receive \( \$ 1,300,000.00 \) in 40 years - Annual Rate of Return: **10.00%** Thus, for the specific investment of \( \$ 3,600.00 \) today to receive \( \$ 10,465.82 \) in 17 years, Keiko will earn an annual rate of return of **6.00%**.