$1)$ An oil company has three different processes that can be used to manufacture different types of gasoline. Each process involves blending oils in the company$$s catalytic cracker. Running process $1$ for an hour costs $$5$ and requires $2$ barrels of crude oil $1$ and $3$ barrels of crude oil $2.$ The output from running process $1$ for an hour is $2$ barrels of gas $1$ and $1$ barrel of gas $2.$ Running process $2$ for an hour costs $$4$ and requires $1$ barrel of crude $1$ and $3$ barrels of crude $2.$ The output from running process $2$ for an hour is $3$ barrels of gas $2.$ Running process $3$ for an hour costs $$1$ and requires $2$ barrels of crude $2$ and $3$ barrels of gas $2.$ The output from running process $3$ for an hour is $2$ barrels of gas $3.$ Each week, $200$ barrels of crude $1,$ at $$2/$barrel$,$ and $300$ barrels of crude $2,$ at $$3/$barrel$,$ may be purchased. All gas produced can be sold at the following per$-$barrel prices: gas $1,$ $$9$; gas $2,$ $$10$; gas $3,$ $$24.$ Assume that only $100$ hours of time on the catalytic cracker are available each week.