An industrial chemical company has three facilities A, B, and C. Each facility produces polyethylene ( PE), polyvinyl chloride ( PVC), and polystyrene ( PS). The table below gives the daily production output ( in metric tons) for each facility: $$\begin{matrix} & A & B & C\cr PE &30 & 80 & 30 \cr PVC &60 & 100 & 80 \cr PS &20 & 60 & 50 \cr \end{matrix}$$

(a) Give vectors a, b, and c that give the daily production amounts at each facility.

Production at A is given by $a=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

Production at B is given by $b=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

Production at C is given by $c=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

(b)

Compute $25 c=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

The entries tell us that in facility C, the company produces tons of PE, tons of PVC, and tons of PS over a period of days.

( c) Determine the combined output from all three facilities over a $3$-week period. ( Note: The facility does not operate on weekends.) Over that period we produce tons of PE, tons of PVC, and tons of PS in the three facilities.

(d) Determine the number of days of production from from each facility required to produce $1060$ tons of PE, $1760$ tons of PVC, and $940$ tons of PS. For that production we need days at facility A, days at facility B, and days at facility C.