The population of a city is modeled by the equation $P(t)=359,767 e^{0.2 t}$ where $t$ is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?
Round your answer to the nearest hundredth of a year (i.e. 2 decimal places).
The population will reach one million in Number years.

Step 1 ：The population of a city is modeled by the equation \(P(t)=359,767 e^{0.2 t}\) where \(t\) is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million? Round your answer to the nearest hundredth of a year (i.e. 2 decimal places).

Step 2 ：The question is asking for the time it will take for the population to reach one million. This means we need to solve the equation \(P(t)=359,767 e^{0.2 t}\) for \(t\) when \(P(t) = 1,000,000\).

Step 3 ：To do this, we can first divide both sides of the equation by 359,767 to isolate the exponential term. Then, we can take the natural logarithm of both sides to solve for \(t\).

Step 4 ：Let \(P_t = 1000000\) and \(P_0 = 359767\).

Step 5 ：Solving for \(t\), we get \(t = 5.11\).

Step 6 ：Final Answer: The population will reach one million in \(\boxed{5.11}\) years.