Question

8) (Application to Population Mixing.) About 6% of a city's population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city (a) Set up a matrix expression that describes this situation, e. find a matrix A such that if x is the city-suburbs population vector for a given year, then Ax is (b) In 2018, there were 9 million residents in the city and 6 million in the suburbs. be the initial population vector. Use your A to estimate the populations in the city and the suburbs two years later, in 2020. Show this the city-suburbs population vector for the following year Let Xo 16 computation by hand, though you may use GeoGebra to ched 9 (c) As time increases off into the distant future, do the city and suburb popula- tions stabilize? If so, estimate those stable populations. If not, what seems to be happening Clearly explain everything and use GeoGebra to support your arguments

Answer 1

Let any times (or year) city population is kpcenel wburtam population is y_0 x_0 = [no: city population y_0: suburbs population] according to question x_1 is born next year X_1 = [X_1 = 94/100 no + 4/100 y_0 y_1 = 6/100 no + 96/100 y_0] (a) x_1 = 94/100 4/100 6/100 96/100] times 0 = 1/100 [94 4 6 96] [x_0 y_0] matrix A = 1/100 [94 4 6 96] (b) in 2018, no = 10, 000, 000 so = 8, 00, 000 in 2019 x_1 = A times 0 = 1/100 [94 4 6 96] [10000 000 8000 000] x_1 = 1/100 [972000000 828000000] = [9720000 8280000] in 2020 times x_2 = Ax_1 = 1/100 [94 4 6 96] [9720000 8280000] \r\n = 1/100 [946800000 853200000] = [9468000 8532000] so in 2020, city population = 9, 468, 000 suburbs population = 8, 532, 000