The rates of change in population for two cities are $P^{\prime}(t)=45$ for Alphaville and $Q^{\prime}(t)=102 e^{0.04 t}$ for Betaburgh, where $t$ is the number of years since 1990 , and $P^{\prime}$ and $Q^{\prime}$ are measured in people per year. In 1990, Alphaville had a population of 5000 , and Betaburgh had a population of 3500 . Answer parts a) through c). a) Determine the population models for both cities. The population model for Alphaville is $\mathrm{P}(\mathrm{t})=45 \mathrm{t}+5000$. The population model for Betaburgh is $Q(t)=2550 e^{.04 t}+950$. b) What were the populations of Alphaville and Betaburgh, to the nearest hundred, in 2000 ? The population of Alphaville in 2000 was people. (Round to the nearest hundred as needed.)

Step 1 ：The population model for Alphaville is \(P(t)=45t+5000\).

Step 2 ：The population model for Betaburgh is \(Q(t)=2550e^{.04t}+950\).

Step 3 ：We substitute \(t=10\) (since 2000 is 10 years after 1990) into the population models to find the populations.

Step 4 ：The population of Alphaville in 2000 is \(P_{2000} = 5450\) people.

Step 5 ：The population of Betaburgh in 2000 is \(Q_{2000} = 4754.152978985239\) people.

Step 6 ：We round these numbers to the nearest hundred.

Step 7 ：The rounded population of Alphaville in 2000 is \(P_{2000_{rounded}} = 5400\) people.

Step 8 ：The rounded population of Betaburgh in 2000 is \(Q_{2000_{rounded}} = 4800\) people.

Step 9 ：Final Answer: The population of Alphaville in 2000 was approximately \(\boxed{5400}\) people and the population of Betaburgh in 2000 was approximately \(\boxed{4800}\) people.