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. d. What annual rate of return will Keiko earn if she invests \( \$ 33,619.78 \) today and receives \( \$ 1,300,000.00 \) in 40 years? \( \% \) (Round to two decimal places.)

To determine the annual rate of return for Keiko's investment, we can use the formula for compound interest, which is: \[ A = P(1 + r)^t \] 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 (as a decimal). - \( t \) is the number of years the money is invested. For Keiko's investment of \( \$ 33,619.78 \) that grows to \( \$ 1,300,000.00 \) over \( 40 \) years, we can rearrange the formula to solve for \( r \): 1. \( 1,300,000 = 33,619.78(1 + r)^{40} \) 2. Divide both sides by \( 33,619.78 \): \[ \frac{1,300,000}{33,619.78} = (1 + r)^{40} \] 3. Calculate \( \frac{1,300,000}{33,619.78} \): \[ \frac{1,300,000}{33,619.78} \approx 38.675 \] 4. Now take the 40th root of both sides to solve for \( r \): \[ 1 + r = (38.675)^{\frac{1}{40}} \] 5. Using a calculator, we find: \[ (38.675)^{\frac{1}{40}} \approx 1.100047 \] Hence, \[ r \approx 1.100047 - 1 \approx 0.100047 \] Now converting \( r \) to a percentage: \[ r \approx 0.100047 \times 100 \approx 10.00\% \] So, Keiko will earn an annual rate of return of approximately **10.00%**. --- Delving into the world of investments, it's essential to understand how powerful compounding can be. Just imagine: if you invest wisely, a little today can turn into a fortune tomorrow! Historically speaking, the stock market has averaged about a 7% return annually over the long haul, which highlights the potential for growth when you allow your money to work for you. As for practical tips, avoid the common pitfall of making emotional investment decisions! Stick to your plan and closely monitor your investments without allowing panic to dictate your actions. Prepare for market fluctuations so you can stay focused on your long-term goals and reap the benefits of patience!