1. **State the problem:** We have an initial bacteria population of 50, which increases by 80% every 20 minutes. We want to find the time it takes for the population to reach 1,200,000. 2. **Formula used:** The population growth can be modeled by the exponential growth formula: $$ P = P_0 \times (1 + r)^n $$ where: - $P$ is the final population, - $P_0$ is the initial population, - $r$ is the growth rate per period (as a decimal), - $n$ is the number of periods. 3. **Identify values:** - $P_0 = 50$ - $P = 1,200,000$ - $r = 0.80$ (80%) - Each period is 20 minutes. 4. **Set up the equation:** $$ 1,200,000 = 50 \times (1 + 0.80)^n = 50 \times 1.8^n $$ 5. **Solve for $n$:** Divide both sides by 50: $$ \frac{1,200,000}{50} = 1.8^n $$ $$ 24,000 = 1.8^n $$ Take the natural logarithm of both sides: $$ \ln(24,000) = \ln(1.8^n) = n \ln(1.8) $$ Solve for $n$: $$ n = \frac{\ln(24,000)}{\ln(1.8)} $$ Calculate values: $$ \ln(24,000) \approx 10.09 $$ $$ \ln(1.8) \approx 0.5878 $$ $$ n \approx \frac{10.09}{0.5878} \approx 17.17 $$ 6. **Calculate total time:** Each period is 20 minutes, so total time $t$ is: $$ t = 17.17 \times 20 = 343.4 \text{ minutes} $$ 7. **Final answer:** It will take approximately **343.4 minutes** for the population to reach 1.2 million bacteria.