DIFFERENCE EQUATIONS A rental care company has a fleet of ab

DIFFERENCE EQUATIONS A rental care company has a fleet of about 500 cars, at three locations. A car rented at one location may be returned to any of the three locations. The various fractions of cars returned to the three locations are shown in the matrix below. Suppose that on Monday thre are 295 cars at the airport (or rented from there), 55 cars at the east side office, and 150 cars in the west side office. What will be the approximate distribution of cars on Wednesday?

Solution

Let the number of cars at the three offices on Tuesday and Wednesday be as under:

Then x₁ = 0.97(295) + 0.05(55) + 0.10(150) = 286+ 3 + 15 = 304

y₁ = 0.00(295) + 0.90(55) + 0.05(150) = 50 + 7 = 57

z₁ = 0.03(295) + 0.05(55) + 0.85(150) = 9 + 3 + 127 = 139

x₂ = 0.97(x₁) + 0.05(y₁) + 0.10(z₁) = 0.97(304) + 0.05(57) + 0.10(139) = 295 + 3 + 14 = 312

y₂ =  0.00(x₁) + 0.90(y₁) + 0.05(z₁) =  0.00(304) + 0.90(57) + 0.05(139) = 51 + 7 = 58

z₂ = 0.03(x₁) + 0.05(y₁) + 0.85(z₁) = 0.03(304) + 0.05(57) + 0.85(139) = 9 + 3 + 118 + 130

Thus the number of cars at the three offices on wednesday will be as under:

Day	Airport	Eastside Office	Westside Office
Monday	x1	y1	z1
Tuesday	x2	y2	z2