The minimum value of the objective function Z | vedclass

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(D) The constraints are $x + y \geq 5$,$x \leq 4$,$y \leq 2$,$x \geq 0$,and $y \geq 0$. To find the feasible region,we identify the intersection point

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$(4, 1)$

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(D) The constraints are $x + y \geq 5$,$x \leq 4$,$y \leq 2$,$x \geq 0$,and $y \geq 0$. To find the feasible region,we identify the intersection points of the lines: $1$. $x + y = 5$ and $x = 4$ gives $4 + y = 5 \implies y = 1$. Point: $(4, 1)$. $2$. $x + y = 5$ and $y = 2$ gives $x + 2 = 5 \implies x = 3$. Point: $(3, 2)$. $3$. $x = 4$ and $y = 2$ gives the point $(4, 2)$. The vertices of the feasible region are $(4, 1)$,$(4, 2)$,and $(3, 2)$. Now,evaluate the objective function $Z = 5x + 8y$ at these vertices: - At $(4, 1)$: $Z = 5(4) + 8(1) = 20 + 8 = 28$. - At $(4, 2)$: $Z = 5(4) + 8(2) = 20 + 16 = 36$. - At $(3, 2)$: $Z = 5(3) + 8(2) = 15 + 16 = 31$. The minimum value is $28$,which occurs at the point $(4, 1)$.

The minimum value of the objective function $Z = 5x + 8y$,subject to the constraints $x + y \geq 5$,$x \leq 4$,$y \leq 2$,$x \geq 0$,and $y \geq 0$,occurs at the point:

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