Question

In a certain region, 6% of a city’s population moves to the surrounding suburbs each year, and 4% of the suburban population moves into the city. A 2019 census measures 10,000,000 residents in the city and 800,000 in the suburbs. (a) Set up a difference equation (of the form ~xn+1 = A~xn) that describes this situation, where ~x0 is the initial population in 2019. (b) Estimate the populations in the city and suburbs two years later, in 2021.

Answer 1

Answer:

Follows are the solution to this question:

Step-by-step explanation:

Following are the differential equation:

$\to P(C,S)t+1 = P(C,S)t+(- 0.06Ct +0.04St, 0.06Ct - 0.04St ) \ \ \ \\\\or \ \ \ \ \ \ \ \\\\ \to (0.94Ct + 0.04St, 0.96St + 0.06Ct)$

In equation:

$\to t = 0, \ as \ 2019\\\\\to Ct = 10,000,000 \\\\\to St = 800,000,000$

$\to P(C, S)1 = (0.94Ct + 0.04St, 0.96St + 0.06Ct)$

                   $= (0.94(10000000) + 0.04(800000), 0.96(800000) + 0.06(10000000))\\\\ = (9,400,000 + 32,000), (768,000 +6000000))\\\\ = (9432000, 1368000)$

$\to P(C, S)2 = (0.94(9432000) + 0.04(1368000), 0.96(1368000) + 0.06(9432000))$

                  $=(8,866,080 + 54,720), (1,313,280 + 565,920)\\\\=(8,920,800, 1,879,200)$