The exponential model $\mathrm{A}=27 e^{0.007 t}$ describes the population, $\mathrm{A}$, of a country in millions, tyears after 2003. Use the model to determine when the population of the country will be 31 million.
The population of the country will be 31 million in (Round to the nearest year as needed.)
Answer
Final Answer: The population of the country will be 31 million in the year \(\boxed{2023}\).
Step 1 ：The exponential model \(A=27 e^{0.007 t}\) describes the population, \(A\), of a country in millions, t years after 2003. We need to determine when the population of the country will be 31 million.
Step 2 ：To find this, we need to solve the equation \(31 = 27e^{0.007t}\) for \(t\).
Step 3 ：This involves taking the natural logarithm of both sides to isolate \(t\).
Step 4 ：By solving the equation, we find that \(t = 20\).
Step 5 ：Since the model starts from the year 2003, we add 20 to 2003 to get the year.
Step 6 ：Final Answer: The population of the country will be 31 million in the year \(\boxed{2023}\).
