After 5.00 pm, tourists will start to arrive in Gillen and they will stay overnight.

As a result, the number of people in Gillen will increase and the television viewing habits of the tourists will also be monitored.

Assume that 50 tourists arrive every hour.

It is expected that 80% of arriving tourists will watch only `C_2` during the hour that they arrive.

The remaining 20% of arriving tourists will not watch television during the hour that they arrive.

Let `W_m` be the state matrix that shows the number of people in each category `m` hours after 5.00 pm on this day.

The recurrence relation that models the change in the television viewing habits of this increasing number of people in Gillen `m` hours after 5.00 pm on this day is shown below.
 

`W_(m + 1) = TW_m + V`
 

where

`{:(quad qquad qquad qquadqquadqquadquadtext(this hour)),(qquadqquadqquad quad \ C_1 qquad quad C_2 qquad \ C_3 quad \ NoTV),(T = [(quad 0.50, 0.05, 0.10, 0.20 quad),(quad 0.10, 0.60, 0.20, 0.20 quad),(quad 0.25, 0.10, 0.50, 0.10 quad),(quad 0.15, 0.25, 0.20, 0.50 quad)]{:(C_1),(C_2),(C_3),(NoTV):}\ text(next hour,) qquad and qquad  W_0 = [(400), (600), (300),(700)]):}`
 

1. Write down matrix `V`.  (1 mark)
2. How many people in Gillen are expected to watch `C_2` at 7.00 pm on this day?  (2 marks)

On Sunday, matrix `V` is used when calculating the expected number of visitors at each location every hour after 10 am. It is assumed that the park will be at its capacity of 2000 visitors for all of Sunday.

Let `L_0` be the state matrix that shows the number of visitors at each location at 10 am on Sunday.

The number of visitors expected at each location at 11 am on Sunday can be determined by the matrix product

 
`{:(qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquad text(this hour)),(qquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquadqquad qquad qquad  \ A qquad quad F qquad \  G \ quad quad W),({:V xx L_0 qquad text(where) qquad L_0 = [(500), (600), (500), (400)]{:(A),(F),(G),(W):}, qquad text(and):} qquad V = [(0.3,0.4,0.6,0.3),(0.1,0.2,0.1,0.2),(0.1,0.2,0.2,0.1),(0.5,0.2,0.1,0.4)]{:(A),(F),(G),(W):}\ text(next hour)):}`
 

1. Safety restrictions require that all four locations have a maximum of 600 visitors.

     

   Which location is expected to have more than 600 visitors at 11 am on Sunday?   (1 mark) 

2. Whenever more than 600 visitors are expected to be at a location on Sunday, the first 600 visitors can stay at that location and all others will be moved directly to Ground World `(G)`.

     

   State matrix `R_n` contains the number of visitors at each location `n` hours after 10 am on Sunday, after the safety restrictions have been enforced.

     

   Matrix `R_1` can be determined from the matrix recurrence relation

     

    
   `qquad qquad qquad R_0 = [(500),(600),(500),(400)]{:(A),(F),(G),(W):}, qquad qquad R_1 = V xx R_0 + B_1`

      

   
   where matrix `B_1` shows the required movement of visitors at 11 am.

1. 1. Determine the matrix `B_1`.  (1 mark)
      
   2. State matrix `R_2` can be determined from the new matrix rule

        

      
      `qquad qquad R_2 = VR_1 + B_2`

        

      
      where matrix `B_2` shows the required movement of visitors at 12 noon.

        

      Determine the state matrix `R_2`.  (1 mark)

Beginning in the year 2021, a new company takes over the maintenance of the 2700 km highway with a new contract.

Each year sections of highway must be graded `(G)`, resurfaced `(R)` or sealed `(S)`.

The remaining highway will need no maintenance `(N)` that year.

Let  `M_n` be the state matrix that shows the highway maintenance schedule of this company for the `n`th year after 2020.

The maintenance schedule for 2020 is shown in matrix  `M_0` below.

For these 2700 km of highway, the matrix recurrence relation shown below can be used to determine the number of kilometres of this highway that will require each type of maintenance from year to year.
 
`qquad qquad M_(n + 1) = TM_n + B`

where

1. Write down the value of `k` in matrix `B`. (1 mark)
2. How many kilometres of highway are expected to be graded `(G)` in the year 2022?  (1 mark)

At a fish farm:

• • young fish (`Y`) may eventually grow into juveniles (`J`) or they may die (`D`)
  •  juveniles (`J`) may eventually grow into adults (`A`) or they may die (`D`)
  • adults (`A`) eventually die (`D`).

The initial state of this population, `F_0`, is shown below.
 

`F_0 = [(50\ 000),(10\ 000),(7000),(0)]{:(Y),(J),(A),(D):}`

Every month, fish are either sold or bought so that the number of young, juvenile and adult fish in the farm remains constant.

The population of fish in the fish farm after `n` months, `F_n`, can be determined by the recurrence rule
 

`F_(n + 1) = [(0.65,0,0,0),(0.25,0.75,0,0),(0,0.20,0.95,0),(0.10,0.05,0.05,1)]\ F_n + B`
 

where `B` is a column matrix that shows the number of young, juvenile and adult fish bought or sold each month and the number of dead fish that are removed.

Each month, the fish farm will

1. sell 1650 adult fish.
2. buy 1750 adult fish.
3. sell 17 500 young fish.
4. buy 50 000 young fish.
5. buy 10 000 juvenile fish.