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.)
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.
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.