For each exercise: It is essential that your graph must explain all the important variables. In other words, the scale of the graph should be selected properly, and all the constraints, the feasible area, all the critical $($corner$)$ points should be SOLVED ALGEBRAICALLY and indicated clearly. The line$($s$)$ also have to be labeled $($colored$)$ clearly.$$ Points will be taken off for sloppiness.$$USE GRAPH PAPER AND COLORS $$ especially when using technology!$$ Answer the question with a complete sentence explaining the meaning of the solution.

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$1.$ Maximize:

C $=3$x $+4$y

Constraints: x $0,$ and y $0$

x $+$ y $10$

$$ x $+$ y $5$

$2$x $+4$y $32$

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Solve the application.$($Be sure to show the Objective function, Constraints, the feasible region, systems of equations, vertices $($CP$),$ and optimal solution.$)$

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$2.$ A manufacturer wants to maximize the profit of two products.$$ Product I yields a profit of $ $1\mathrm{.}50$ per unit, and product II yields a profit of $$2\mathrm{.}00$ per unit.$$ Market tests and available resources have indicated the following constraints:

The combined production level should not exceed $1200$ units per month.

The demand for product II is no more than half the demand for product I.

The production level of product I is less than or equal to $600$ units plus three times the production level of product II$.$