1. **State the problem:** We are given the exponential population model $$A = 663.8e^{0.024t}$$ where $A$ is the population in millions and $t$ is the number of years after 2003. We need to find the value of $t$ when the population $A$ reaches 1331 million. 2. **Write the equation to solve:** Set $$A = 1331$$ and solve for $t$: $$1331 = 663.8e^{0.024t}$$ 3. **Isolate the exponential term:** Divide both sides by 663.8: $$\frac{1331}{663.8} = e^{0.024t}$$ Calculate the left side: $$2.005 = e^{0.024t}$$ 4. **Take the natural logarithm of both sides:** $$\ln(2.005) = \ln\left(e^{0.024t}\right)$$ Using the property $\ln(e^x) = x$: $$\ln(2.005) = 0.024t$$ 5. **Solve for $t$:** $$t = \frac{\ln(2.005)}{0.024}$$ Calculate $\ln(2.005)$: $$\ln(2.005) \approx 0.695$$ So, $$t = \frac{0.695}{0.024} \approx 28.96$$ 6. **Interpret the result:** Since $t$ is years after 2003, the population will reach 1331 million approximately 29 years after 2003. **Final answer:** The population will be 1331 million in the year $$2003 + 29 = 2032$$. This means the population reaches 1331 million around the year 2032.