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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
Related questions
Question
100%
![**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{
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning