MATH152 April 2016
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Question B 04 (d)
Suppose in the year 2020, 50 million people live in cities and 50 million in the suburbs.
Every year, 10% of city residents move to the suburbs and 20% of the residents of the suburbs move to cities.
(d) Assuming the overall population does not change (i.e., remains at 100 million), how many people will be living in the suburbs far in the future?
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
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We should compute for large In other words, we need to find
In order to do this, we consider "Matrix Diagonalization" to write and therefore
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We first use the Matrix diagonalization method to write the matrix in the following form:
- where and
From part (a) and (c), we have and Also, and So, we have and therefore,
We finally get that
which means for any time we have
Now we are ready to find for large by computing So,
Now by applying this matrix on our initial population we get
Therefore, million people will be living in the suburbs far in the future.
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