A toy company manufactures two types of dolls | vedclass

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(A) Let $x$ and $y$ be the number of dolls of type $A$ and $B$ respectively that are produced per week.The given problem can be formulated as follows:

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$800$ dolls of type $A$ and $400$ dolls of type $B$

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(A) Let $x$ and $y$ be the number of dolls of type $A$ and $B$ respectively that are produced per week.The given problem can be formulated as follows:Maximize $z = 12x + 16y$ .....$(1)$Subject to the constraints:$x + y \leq 1200$ .....$(2)$$y \leq \frac{x}{2} \Rightarrow x \geq 2y$ .....$(3)$$x - 3y \leq 600$ .....$(4)$$x, y \geq 0$ .....$(5)$The feasible region is determined by the system of constraints. The corner points of the feasible region are $O(0,0)$,$A(600,0)$,$B(1050,150)$,and $C(800,400)$.The values of $z$ at these corner points are as follows:      Corner point    $z = 12x + 16y$        $O(0,0)$    $0$        $A(600,0)$    $7200$        $B(1050,150)$    $12(1050) + 16(150) = 12600 + 2400 = 15000$        $C(800,400)$    $12(800) + 16(400) = 9600 + 6400 = 16000$  The maximum value of $z$ is $16000$ at $(800, 400)$.Thus,$800$ dolls of type $A$ and $400$ dolls of type $B$ should be produced weekly to maximize the profit.

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