3 A study conducted by the State Department of Agriculture i

3. A study conducted by the State Department of Agriculture in a Sun-Belt state reveals an land within the state and rd urbanization of the farmland within the state. In 1990 48% of the was used for agricultural purposes(A), 18% had been urbanized U) the remaining 34% was neither (N) agricultural nor urban. It was found that each year 3% of the agricultural land has been converted to urban land, 196 has been used for other purposes, and the remaining 96% is stayed agricultural of the urban land, 196 has been used for other purposes and the remaining 99% stayed urban. of the land that was neither agricultural nor urban, 4% has been converted to agricultural land and 2% has been urbanized, the remaining 94% was left as neither. . . a. Write the transition matrix for this problem. b. Find the proportion of agricultural land, urban land and land used for other purposes in this state one year later (1991). Label your answers. c. Find the proportion of agricultural land, urban land and land used for other purposes in this state in the year 2000. Label your answers.

Solution

let P be transition matrix;

bi = [x;y;z] ; where x is percentage of Agriculture ,y  is percentage of urban land , z  is percentage of other uses.

given condition

P = [0.96 0 0.04 ; 0.03 0.99 0.02 ; 0.01 0.01 0.94];

P =

0.9600 0 0.0400
0.0300 0.9900 0.0200
0.0100 0.0100 0.9400

b0 = [0.48;0.18;0.34];

b0 =

0.4800
0.1800
0.3400 {b0 in year 1990

b) so in year 1991

b1 = P*b0

b1 =

0.4744
0.1994
0.3262

c) in year 2010

b10 = (P^10)*b0;

b10

b10 =

0.4145
0.3472
0.2383