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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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