Question

The transit authority has made a study that analyzed the percentages of commuters using public transportation and personal cars. They found that 80% of the commuters using public transportation in a given year will keep using public transportation in the following year, and that 30% of those driving personal cars in a given year will switch to public transportation the following year. Assuming that the population remains constant and the results of the study hold in subsequent years, find the percentages of the commuters using public transportation and personal cars in the long run.

Answer 1

Answer:

Step-by-step explanation:

given that the transit authority has made a study that analyzed the percentages of commuters using public transportation and personal cars

80% will keep same as public transportions and 20% will switch to personal cars.

Similarly 30% of personal cars will switch to public transportation

Let A be public trans. and B be personal cars.

Next year we will have A = 0.8A+0.3B

Similarly next B = 0.2A+0.7B

Transition matrix is

$\left[\begin{array}{ccc}0.8&0.3\\0.2&0.7\end{array}\right]$

After n years we would have

(A  B) * C^n where c is the transition matrix

In 2  year it would be$\left[\begin{array}{ccc}0.7&0.45\\0.3&0.55\end{array}\right]$

A' = 0.7A+0.45B and

B' = 0.3A+0.55B

After 4 years

A' =0.625A+0.5625B

B' = 0.375A+0.4375B

When this is stationary we would have

A B =(A  B) * C

A = 0.8A+0.3B

B = 0.2A+0.7B

Or 0.2A = 0.3B

Thus when A and B are in the ratioi 2:3 there will not be any change.

In the long run, 40% would be using public and 60% personal cars