The exponential model $A=106.1 e^{0.02 t}$ describes the population, $A$, of a country in millions, $t$ years after 2003 . Use the model to determine when the population of the country will be 155 million.
The population of the country will be 155 million in (Round to the nearest year as needed)

Step 1 ：The exponential model \(A=106.1 e^{0.02 t}\) describes the population, \(A\), of a country in millions, \(t\) years after 2003. We are asked to determine when the population of the country will be 155 million.

Step 2 ：To find this, we need to solve the equation \(155 = 106.1 e^{0.02 t}\) for \(t\). This is a logarithmic equation, so we'll need to use the properties of logarithms to solve it.

Step 3 ：First, divide both sides of the equation by 106.1 to isolate \(e^{0.02 t}\) on one side of the equation. This gives us \(\frac{155}{106.1} = e^{0.02 t}\).

Step 4 ：Next, take the natural logarithm of both sides of the equation to get rid of the exponential on the right side. This gives us \(\ln(\frac{155}{106.1}) = 0.02 t\).

Step 5 ：Finally, divide both sides of the equation by 0.02 to solve for \(t\). This gives us \(t = \frac{\ln(\frac{155}{106.1})}{0.02}\).

Step 6 ：Calculating the right side of the equation gives us \(t \approx 19\).

Step 7 ：Since \(t\) represents the number of years after 2003, we add 19 to 2003 to find the year in which the population will be 155 million.

Step 8 ：Final Answer: The population of the country will be 155 million in the year \(\boxed{2022}\).