$2.$ The Mill Mountain Coffee Shop blends coffee on the premises for its customers. It sells three basic blends in $1-$pound bags, Special, Mountain Dark, and Mill Regular. It uses four different types of coffee to produce the blends $-$ Brazilian, mocha, Colombian, and mild. The shop has the following blend recipe requirements:

The cost of Brazilian coffee is $\backslash (\backslash$$ $2\mathrm{.}00\backslash )$ per pound, the cost of mocha is $\backslash (\backslash$$ $2\mathrm{.}75\backslash )$ per pound, the cost of Colombian is $\backslash (\backslash$$ $2\mathrm{.}90\backslash )$ per pound, and the cost of mild is $\backslash (\backslash$$ $1\mathrm{.}70\backslash )$ per pound. The shop has $110$ pounds of Brazilian coffee, $70$ pounds of mocha, $80$ pounds of Colombian, and $150$ pounds of mild coffee available per week. The shop wants to know the amount of each blend it should prepare each week to maximize profit.

$($a$)$ Define the decision variables

$($b$)$ Write the objective function

$($c$)(6$Pts$)$ List all the constraints.