Problem 13. (2+3+2=7 points) Every year, 10% of the population of Richmond moves
ID: 3117447 • Letter: P
Question
Problem 13. (2+3+2=7 points) Every year, 10% of the population of Richmond moves to Vancouver, and 20% of the population of Vancouver moves to Richmond. Assume that there are no other (a) If the total population of the two cities is 3 million, what are the populations (b) Assuming that in the current year, the population of Vancouver is 2 million, effects on the populations of these two cities. of the two cities in the long run? and that of Richmond is 1 million. Find precise formulas for the values of the poplulations of the two cities after n years. (c) Continuing with the assumptions of (b), after how many years will the popu- lation of Richmond for the first time surpass the population of Vancouver?Explanation / Answer
Let current population of Vancouver in ith year from now be denoted by V(i) and that of Richmond by R(i). So, given information can be represented mathematically as
V(i+1) = (1 - 20%) V(i) + 10% R(i)
R(i+1) = 20% V(i) + (1 - 10%) R(i)
or
V(i+1) = 0.8 V(i) + 0.1 R(i)
R(i+1) = 0.2 V(i) + 0.9 R(i)
or
[V(i+1) R(i+1)]T = [[0.8 0.1], [0.2 0.9]] x [V(i) R(i)]T
or
[V(i+1) R(i+1)]T = A x [V(i) R(i)]T
where
A = [[0.8 0.1], [0.2 0.9]]
We are representing 2x2 matrix here as [Row1, Row2], and a column vector in Row transpose form.
(a) Since there are no other effects on the populations of the two cities other than people moving from one to another, total population of two cities would remain constant. This can also be verified mathematically.
V(i+1) + R(i+1) = (0.8 + 0.2) V(i) + (0.1 + 0.9) R(i) = V(i) + R(i)
Hence, if the total population of the two cities is 3 million, it would remain same even in the long run.
(b)
[V(i+2) R(i+2)]T = A x [V(i+1) R(i+1)]T
= A x A x [V(i) R(i)]T
= (A)2 x [V(i) R(i)]T
Hence,
[V(i+n) R(i+n)]T = (A)n x [V(i) R(i)]T
Setting i to 0, we get
[V(n) R(n)]T = (A)n x [V(0) R(0)]T
Given the figures for current year (i=0),
V(0) = the population of Vancouver = 2 million
R(0) = the population of Richmond = 1 million,
the poplulations of these two cities after n years can be expressed precisely as
[V(n) R(n)]T = (A)n x [2*10^6 1*10^6]T
where matrix A = [[0.8 0.1], [0.2 0.9]]
(c) Continuing with the assumptions of (b), let us compute the values of Vancouver and Richmond populations for few years.
n | V(n) | R(n)
0 | 2000000 | 1000000
1 | 1700000 | 1300000
2 | 1490000 | 1510000
So, we notice that after two years the population of Richmond will for the rst time surpass the population of Vancouver.