The exponential model $\mathrm{A}=481.3 e^{0.023 t}$ describes the population, $\mathrm{A}$, of a country in millions, $\mathrm{t}$ years after 2003. Use the model to determine when the population of the country will be 620 million.

Step 1 ：Given the exponential model \(A = 481.3e^{0.023t}\), where \(A\) is the population in millions and \(t\) is the number of years after 2003.

Step 2 ：We want to find when the population will be 620 million, so we set \(A = 620\) and solve for \(t\).

Step 3 ：First, we divide both sides of the equation by 481.3 to isolate the exponential term: \(\frac{620}{481.3} = e^{0.023t}\).

Step 4 ：Next, we take the natural logarithm of both sides to remove the exponential: \(\ln(\frac{620}{481.3}) = 0.023t\).

Step 5 ：Finally, we divide both sides by 0.023 to solve for \(t\): \(t = \frac{\ln(\frac{620}{481.3})}{0.023}\).

Step 6 ：Calculating the right side gives \(t \approx 11.01\).

Step 7 ：Since \(t\) represents the number of years after 2003, we add 2003 to get the year: 2003 + 11 = 2014.

Step 8 ：Final Answer: The population of the country will be 620 million approximately \(\boxed{11}\) years after 2003, which is in the year 2014.