Question

[M] Budget Rent A car in Wichit, Kanas, 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 fraction of cars returned to the three locations are shown in the matrix below. Suppose that on Monday there are 295 cars at the airport (or rented from there), 55 cars at the east side office, and 150 cars at the west side of office. What will be the approximate distribution of cars on Wednesday?

Cars Rented Form			Returned To
Airport	East	West
0.97	0.05	0.10	Airport
0.00	0.90	0.05	East
0.03	0.05	0.85	West

Short answer

\(\left[ {\begin{array}{*{20}{c}}{312}\\{58}\\{130}\end{array}} \right]\)

Step-by-step solution

Form the difference equation

The matrix \(M = \left[ {\begin{array}{*{20}{c}}{0.97}&{0.05}&{0.10}\\{0.00}&{0.90}&{0.05}\\{0.03}&{0.05}&{0.85}\end{array}} \right]\) and the matrix \({{\bf{x}}_k}\) represent the car available at the airport on a particular day.

Let \(k = 0\) represent Monday. Then, by the difference equation, \({{\bf{x}}_{k + 1}} = M{{\bf{x}}_k}\).

\({{\bf{x}}_0} = \left[ {\begin{array}{*{20}{c}}{295}\\{55}\\{150}\end{array}} \right]\)

Calculate the cars available on Tuesday

For \(k = 0\), by the equation \({{\bf{x}}_{k + 1}} = M{{\bf{x}}_k}\),

\(\begin{array}{c}{{\bf{x}}_1} = \left[ {\begin{array}{*{20}{c}}{0.97}&{0.05}&{0.10}\\{0.00}&{0.90}&{0.05}\\{0.03}&{0.05}&{0.85}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{295}\\{55}\\{150}\end{array}} \right]\\ \approx \left[ {\begin{array}{*{20}{c}}{304}\\{57}\\{139}\end{array}} \right].\end{array}\)

Calculate the cars available on Wednesday

For \(k = 1\), by the equation \({{\bf{x}}_{k + 1}} = M{{\bf{x}}_k}\).

\(\begin{array}{c}{{\bf{x}}_2} = \left[ {\begin{array}{*{20}{c}}{0.97}&{0.05}&{0.10}\\{0.00}&{0.90}&{0.05}\\{0.03}&{0.05}&{0.85}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{304}\\{57}\\{139}\end{array}} \right]\\ \approx \left[ {\begin{array}{*{20}{c}}{312}\\{58}\\{130}\end{array}} \right].\end{array}\)

So, the matrix \(\left[ {\begin{array}{*{20}{c}}{312}\\{58}\\{130}\end{array}} \right]\) represents the cars at three locations on Wednesday.