Questions: The rates of change in population for two cities are P'(t)=42 for Alphaville and Q'(t)=105 e^(0.06 t) for Betaburgh, where t is the number of years since 1990, and P' and Q' are measured in people per year. In 1990, Alphaville had a population of 5500, and Betaburgh had a population of 3000. Answer parts a) through c). b) What were the populations of Alphaville and Betaburgh, to the nearest hundred, in 2000? The population of Alphaville in 2000 was 5900 people. (Round to the nearest hundred as needed.) The population of Betaburgh in 2000 was 4400 people. (Round to the nearest hundred as needed.) c) Sketch the graph of each city's population model and estimate the year in which the two cities have the same population. Choose the correct graph below. A. B. C. D. The two cities have the same population in the year . (Round to the nearest year as needed.)

The rates of change in population for two cities are P'(t)=42 for Alphaville and Q'(t)=105 e^(0.06 t) for Betaburgh, where t is the number of years since 1990, and P' and Q' are measured in people per year. In 1990, Alphaville had a population of 5500, and Betaburgh had a population of 3000. Answer parts a) through c).
b) What were the populations of Alphaville and Betaburgh, to the nearest hundred, in 2000?

The population of Alphaville in 2000 was 5900 people. (Round to the nearest hundred as needed.) The population of Betaburgh in 2000 was 4400 people. (Round to the nearest hundred as needed.)
c) Sketch the graph of each city's population model and estimate the year in which the two cities have the same population. Choose the correct graph below. A. B. C. D.

The two cities have the same population in the year . (Round to the nearest year as needed.)

Transcript text: The rates of change in population for two cities are $P^{\prime}(t)=42$ for Alphaville and $Q^{\prime}(\mathrm{t})=105 e^{0.06 t}$ 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 5500, and Betaburgh had a population of 3000 . Answer parts a) through c). b) What were the populations of Alphaville and Betaburgh, to the nearest hundred, in 2000? The population of Alphaville in 2000 was 5900 people. (Round to the nearest hundred as needed.) The population of Betaburgh in 2000 was 4400 people. (Round to the nearest hundred as needed.) c) Sketch the graph of each city's population model and estimate the year in which the two cities have the same population. Choose the correct graph below. A. B. C. D. The two cities have the same population in the year $\square$ . (Round to the nearest year as needed.)

Solution

Solution Steps

Step 1: Determine the population model for Alphaville

The rate of change in population for Alphaville is given by $ P'(t) = 42 $. Integrating this with respect to $ t $, we get: \[ P(t) = 42t + C \] Given that in 1990 (when $ t = 0 $), the population $ P(0) = 5500 $: \[ 5500 = 42(0) + C \] \[ C = 5500 \] Thus, the population model for Alphaville is: \[ P(t) = 42t + 5500 \]

Step 2: Determine the population model for Betaburgh

The rate of change in population for Betaburgh is given by $ Q'(t) = 105e^{0.061t} $. Integrating this with respect to $ t $, we get: \[ Q(t) = \int 105e^{0.061t} \, dt \] \[ Q(t) = \frac{105}{0.061} e^{0.061t} + C \] \[ Q(t) = 1721.31 e^{0.061t} + C \] Given that in 1990 (when $ t = 0 $), the population $ Q(0) = 3000 $: \[ 3000 = 1721.31 e^{0} + C \] \[ 3000 = 1721.31 + C \] \[ C = 1278.69 \] Thus, the population model for Betaburgh is: \[ Q(t) = 1721.31 e^{0.061t} + 1278.69 \]

Step 3: Calculate the populations in 2000

For Alphaville in 2000 ($ t = 10 $): \[ P(10) = 42(10) + 5500 \] \[ P(10) = 420 + 5500 \] \[ P(10) = 5920 \] Rounding to the nearest hundred: \[ P(10) \approx 5900 \]

For Betaburgh in 2000 ($ t = 10 $): \[ Q(10) = 1721.31 e^{0.061(10)} + 1278.69 \] \[ Q(10) = 1721.31 e^{0.61} + 1278.69 \] \[ Q(10) \approx 1721.31 \times 1.841 + 1278.69 \] \[ Q(10) \approx 3168.91 + 1278.69 \] \[ Q(10) \approx 4447.60 \] Rounding to the nearest hundred: \[ Q(10) \approx 4400 \]