Waiting period. Jamal is waiting to be a millionaire. He wants to know how long he must wait if a. he invests \( \$ 27,523.17 \) at \( 20 \% \) today? b. he invests \( \$ 54,955.85 \) at \( 13 \% \) today? c. he invests \( \$ 126,545.56 \) at \( 11 \% \) today? d. he invests \( \$ 284,998.58 \) at \( 7 \% \) today? a. How long will Jamal have to wait to become a millionaire if he invests \( \$ 27,523.17 \) at \( 20 \% \) today? \( \square \) years (Round to the nearest whole number.)

To determine how long Jamal must wait to become a millionaire with his investments, we can use the formula for compound interest, which is: \[ A = P(1 + r)^t \] where: - \( A \) is the amount of money accumulated after \( t \) years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( t \) is the number of years the money is invested. We're looking to solve for \( t \), which can be done by rearranging the formula to: \[ t = \frac{\log(A/P)}{\log(1 + r)} \] Now, let’s calculate the waiting periods for each investment: ### a. \$27,523.17 at 20% 1. \( P = 27,523.17 \) 2. \( A = 1,000,000 \) (to be a millionaire) 3. \( r = 0.20 \) \[ t = \frac{\log(1,000,000 / 27,523.17)}{\log(1 + 0.20)} \approx \frac{\log(36.4)}{\log(1.20)} \approx \frac{1.5607}{0.07918} \approx 19.7 \] Rounded to the nearest whole number, Jamal will have to wait approximately **20 years**. ### b. \$54,955.85 at 13% 1. \( P = 54,955.85 \) 2. \( A = 1,000,000 \) 3. \( r = 0.13 \) \[ t = \frac{\log(1,000,000 / 54,955.85)}{\log(1 + 0.13)} \approx \frac{\log(18.2)}{\log(1.13)} \approx \frac{1.2608}{0.05407} \approx 23.3 \] Rounded to the nearest whole number, it would take approximately **23 years**. ### c. \$126,545.56 at 11% 1. \( P = 126,545.56 \) 2. \( A = 1,000,000 \) 3. \( r = 0.11 \) \[ t = \frac{\log(1,000,000 / 126,545.56)}{\log(1 + 0.11)} \approx \frac{\log(7.9)}{\log(1.11)} \approx \frac{0.8962}{0.04139} \approx 21.7 \] Rounded to the nearest whole number, he will wait about **22 years**. ### d. \$284,998.58 at 7% 1. \( P = 284,998.58 \) 2. \( A = 1,000,000 \) 3. \( r = 0.07 \) \[ t = \frac{\log(1,000,000 / 284,998.58)}{\log(1 + 0.07)} \approx \frac{\log(3.5)}{\log(1.07)} \approx \frac{0.5441}{0.02963} \approx 18.4 \] Rounded to the nearest whole number, this means he’ll have to wait approximately **18 years**. So here's a quick summary: - 20 years for \$27,523.17 at 20% - 23 years for \$54,955.85 at 13% - 22 years for \$126,545.56 at 11% - 18 years for \$284,998.58 at 7%. Time to start investing and watching that money grow!