###### Exercise12

Suppose that a city is starting a bicycle sharing program with bicycles at locations $$B$$ and $$C\text{.}$$ Bicycles that are rented at one location may be returned to either location at the end of the day. Over time, the city finds that 80% of bicycles rented at location $$B$$ are returned to $$B$$ with the other 20% returned to $$C\text{.}$$ Similarly, 50% of bicycles rented at location $$C$$ are returned to $$B$$ and 50% to $$C\text{.}$$

To keep track of the bicycles, we form a vector

\begin{equation*} \xvec_k = \twovec{B_k}{C_k} \end{equation*}

where $$B_k$$ is the number of bicycles at location $$B$$ at the beginning of day $$k$$ and $$C_k$$ is the number of bicycles at $$C\text{.}$$ The information above tells us

\begin{equation*} \xvec_{k+1} = A\xvec_k \end{equation*}

where

\begin{equation*} A = \left[\begin{array}{rr} 0.8 \amp 0.5 \\ 0.2 \amp 0.5 \\ \end{array}\right]\text{.} \end{equation*}

1. Let's check that this makes sense.

1. Suppose that there are 1000 bicycles at location $$B$$ and none at $$C$$ on day 1. This means we have $$\xvec_1 = \twovec{1000}{0}\text{.}$$ Find the number of bicycles at both locations on day 2 by evaluating $$\xvec_2 = A\xvec_1\text{.}$$

2. Suppose that there are 1000 bicycles at location $$C$$ and none at $$B$$ on day 1. Form the vector $$\xvec_1$$ and determine the number of bicycles at the two locations the next day by finding $$\xvec_2 = A\xvec_1\text{.}$$

2. Suppose that one day there are 1050 bicycles at location $$B$$ and 450 at location $$C\text{.}$$ How many bicycles were there at each location the previous day?

3. Suppose that there are 500 bicycles at location $$B$$ and 500 at location $$C$$ on Monday. How many bicycles are there at the two locations on Tuesday? on Wednesday? on Thursday?

in-context