a. In 2000 , the population of a country was approximately 6.17 million and by 2040 it is projected to grow to 9 million. Use the exponential growth model $\mathrm{A}=\mathrm{A}_{0} e^{\mathrm{kt}}$, in which $\mathrm{t}$ is the number of years after 2000 and $\mathrm{A}_{0}$ is in millions, to find an exponential growth function that models the data. By which year will the population be 13 million? a. The exponential growth function that models the data is $A=$ (Simplify your answer. Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.) b. The country's population will be 13 million in the year (Round to the nearest year as needed.)

Step 1 ：Given that in the year 2000, the population of a country was approximately 6.17 million and by 2040 it is projected to grow to 9 million. We are to use the exponential growth model \(A=A_{0} e^{kt}\), where \(t\) is the number of years after 2000 and \(A_{0}\) is in millions, to find an exponential growth function that models the data.

Step 2 ：First, we need to find the value of \(k\) in the exponential growth model. We know that \(A_{0}\) is 6.17 (the population in 2000), \(A\) is 9 (the population in 2040), and \(t\) is 40 (the number of years from 2000 to 2040). We can substitute these values into the equation and solve for \(k\).

Step 3 ：Having found the value of \(k\) to be approximately 0.00944, we can substitute it back into the equation to get the exponential growth function that models the data, which is \(A = 6.17e^{0.00944t}\).

Step 4 ：Then, to find the year when the population will be 13 million, we can set \(A\) to 13 and solve for \(t\). Since \(t\) is the number of years after 2000, we need to add 2000 to \(t\) to get the actual year.

Step 5 ：Upon solving, we find that \(t\) is approximately 78.96. Adding 2000 to this gives us the year 2079.

Step 6 ：Final Answer: The exponential growth function that models the data is \(A = 6.17e^{0.00944t}\). The country's population will be 13 million in the year \(\boxed{2079}\).