Provide a linear program for the following problem. Don$$t worry that the solution to the linear program might not be integers:

A factory produces Aluminium plates of a standard width of $3$ meters. However, customers want to buy plates of shorter widths. So$,$ the factory has to cut original $3$m plates. One $3$m plate can be cut, for instance, into two plates of $93$cm width, one plate of width $108$cm$,$ and a remainder of $6$cm $($which goes to waste$).$ Suppose that we have the following orders:

$97$ plates of width $135$cm $610$ plates of width $108$cm $395$ plates of width $94$cm $211$ plates of width $43$cm$.$

What is the smallest number of $3$m plates that have to be cut in order to satisfy this order? How should they be cut? $($Hint: there are $12$ different ways of cutting a plate into useful subpieces; you$$ll need to figure them all out.$)$

Do not try to solve the linear program, just present it$.$ Also, suppose that one solves this linear program $$ how can one then answer the two questions in the previous paragraph?