1. **State the problem:** We want to find the time $t$ it takes for a population to grow from 500 to 1200 at an annual growth rate of 12%. 2. **Formula used:** The population growth can be modeled by the exponential growth formula $$P(t) = P_0 (1 + r)^t$$ where $P_0$ is the initial population, $r$ is the growth rate per year, and $t$ is the time in years. 3. **Identify values:** Here, $P_0 = 500$, $P(t) = 1200$, and $r = 0.12$. 4. **Set up the equation:** $$1200 = 500 (1 + 0.12)^t = 500 (1.12)^t$$ 5. **Divide both sides by 500:** $$\frac{1200}{500} = (1.12)^t \implies 2.4 = (1.12)^t$$ 6. **Take the natural logarithm of both sides:** $$\ln(2.4) = \ln((1.12)^t) = t \ln(1.12)$$ 7. **Solve for $t$:** $$t = \frac{\ln(2.4)}{\ln(1.12)}$$ 8. **Calculate values:** $$\ln(2.4) \approx 0.8755, \quad \ln(1.12) \approx 0.1133$$ 9. **Final calculation:** $$t \approx \frac{0.8755}{0.1133} \approx 7.73$$ 10. **Interpretation:** This means it takes approximately 7.73 years for the population to grow from 500 to 1200 at 12% per year. **Note:** The answer given as 19 years seems inconsistent with the 12% growth rate and these values. If the problem intended continuous growth or a different rate, please clarify. For the given data and formula, the time is about 7.73 years.