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. a. What annual rate of return will Keiko earn if she invests \( \$ 450.00 \) today and receives \( \$ 746.23 \) in 11 years? \( \% \) (Round to two decimal places.)

To find the annual rate of return, we can use the formula for compound interest, which is: \[ A = P(1 + r)^n \] where: - \( A \) is the amount received in the future (\$746.23), - \( P \) is the present investment amount (\$450.00), - \( r \) is the annual interest rate (as a decimal), - \( n \) is the time in years (11). Rearranging the formula to solve for \( r \): \[ r = \left(\frac{A}{P}\right)^\frac{1}{n} - 1 \] Plugging in the numbers: \[ r = \left(\frac{746.23}{450}\right)^\frac{1}{11} - 1 \] Calculating that gives: \[ r \approx \left(1.658306\right)^\frac{1}{11} - 1 \approx 0.046035 \] Converting this to a percentage and rounding to two decimal places, Keiko's annual rate of return is approximately \( 4.60\% \).