12. In a certain region about 5% of a city's population moves to the surrounding suburbs each year, and about 3% of the suburban population moves into the city. In 2018, there were 2000000 residents in the city and 750000 in the suburbs. Write down a migration matrix that describes this situation and use it to estimate the expected populations in the city and in the suburbs in 2020.
12. In a certain region about 5% of a city's population moves to the surrounding suburbs each year, and about 3% of the suburban population moves into the city. In 2018, there were 2000000 residents in the city and 750000 in the suburbs. Write down a migration matrix that describes this situation and use it to estimate the expected populations in the city and in the suburbs in 2020.
Elementary Geometry for College Students
6th Edition
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Daniel C. Alexander, Geralyn M. Koeberlein
ChapterA: Appendices
SectionA.1: Algebraic Expressions
Problem 2E
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![**Problem 12: Population Migration Analysis**
In a certain region, about 5% of a city's population moves to the surrounding suburbs each year, and about 3% of the suburban population moves into the city. In 2018, there were 200,000 residents in the city and 75,000 in the suburbs. Write down a migration matrix that describes this situation and use it to estimate the expected populations in the city and in the suburbs in 2020.
### Explanation
To model this scenario, we can use a migration matrix to represent the movement of populations between the city and the suburbs. Let \( C_t \) and \( S_t \) represent the populations of the city and suburbs, respectively, at time \( t \). Then the migration between the city and suburbs can be represented as:
\[ \begin{bmatrix}
C_{t+1} \\
S_{t+1}
\end{bmatrix}
=
\begin{bmatrix}
0.95 & 0.03 \\
0.05 & 0.97
\end{bmatrix}
\begin{bmatrix}
C_t \\
S_t
\end{bmatrix} \]
### Initial Conditions
For the year 2018:
\[ \begin{bmatrix}
C_{2018} \\
S_{2018}
\end{bmatrix}
=
\begin{bmatrix}
200000 \\
75000
\end{bmatrix} \]
### Estimating Populations in 2020
We can calculate the populations in subsequent years using the migration matrix. First, calculate the populations for 2019:
\[ \begin{bmatrix}
C_{2019} \\
S_{2019}
\end{bmatrix}
=
\begin{bmatrix}
0.95 & 0.03 \\
0.05 & 0.97
\end{bmatrix}
\begin{bmatrix}
200000 \\
75000
\end{bmatrix}
=
\begin{bmatrix}
0.95 \times 200000 + 0.03 \times 75000 \\
0.05 \times 200000 + 0.97 \times 75000
\end{bmatrix} =
\begin{bmatrix}
197250 \\
778750
\end{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12a9a3e9-d100-4e80-b765-c2c582e29977%2F80d79bdd-e2d1-4225-8ac1-b60f36e64051%2Fmtj6baj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 12: Population Migration Analysis**
In a certain region, about 5% of a city's population moves to the surrounding suburbs each year, and about 3% of the suburban population moves into the city. In 2018, there were 200,000 residents in the city and 75,000 in the suburbs. Write down a migration matrix that describes this situation and use it to estimate the expected populations in the city and in the suburbs in 2020.
### Explanation
To model this scenario, we can use a migration matrix to represent the movement of populations between the city and the suburbs. Let \( C_t \) and \( S_t \) represent the populations of the city and suburbs, respectively, at time \( t \). Then the migration between the city and suburbs can be represented as:
\[ \begin{bmatrix}
C_{t+1} \\
S_{t+1}
\end{bmatrix}
=
\begin{bmatrix}
0.95 & 0.03 \\
0.05 & 0.97
\end{bmatrix}
\begin{bmatrix}
C_t \\
S_t
\end{bmatrix} \]
### Initial Conditions
For the year 2018:
\[ \begin{bmatrix}
C_{2018} \\
S_{2018}
\end{bmatrix}
=
\begin{bmatrix}
200000 \\
75000
\end{bmatrix} \]
### Estimating Populations in 2020
We can calculate the populations in subsequent years using the migration matrix. First, calculate the populations for 2019:
\[ \begin{bmatrix}
C_{2019} \\
S_{2019}
\end{bmatrix}
=
\begin{bmatrix}
0.95 & 0.03 \\
0.05 & 0.97
\end{bmatrix}
\begin{bmatrix}
200000 \\
75000
\end{bmatrix}
=
\begin{bmatrix}
0.95 \times 200000 + 0.03 \times 75000 \\
0.05 \times 200000 + 0.97 \times 75000
\end{bmatrix} =
\begin{bmatrix}
197250 \\
778750
\end{
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