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First, we need to graph the inequalities $2x+y\ge1000$ and $x+2y\ge800$, along with the non-negativity constraints $x,y\ge0$.
Convert the inequalities to equations to find the boundary lines: $2x+y=1000$ and $x+2y=800$.
For $2x+y=1000$: If $x=0$, then $y=1000$. Point: $(0, 1000)$ If $y=0$, then $2x=1000 \implies x=500$. Point: $(500, 0)$ For $x+2y=800$: If $x=0$, then $2y=800 \implies y=400$. Point: $(0, 400)$ If $y=0$, then $x=800$. Point: $(800, 0)$
Plot the lines $2x+y=1000$ and $x+2y=800$ on the graph. Since the inequalities are $\ge$, shade the region above each line.
The feasible region is the intersection of the shaded regions, which is unbounded. The corner points of the feasible region are the intersection of the lines and the axes.
Solve the system of equations: $2x+y=1000$ $x+2y=800$ Multiply the second equation by 2: $2x+4y=1600$. Subtract the first equation from the modified second equation: $(2x+4y) - (2x+y) = 1600 - 1000$ $3y = 600$ $y = 200$ Substitute $y=200$ into $x+2y=800$: $x+2(200)=800$ $x+400=800$ $x=400$ The intersection point is $(400, 200)$.
The corner points of the feasible region are $(0, 1000)$, $(400, 200)$, and $(800, 0)$.
Evaluate the objective function $Z=x+4y$ at each corner point: At $(0, 1000)$: $Z = 0 + 4(1000) = 4000$ At $(400, 200)$: $Z = 400 + 4(200) = 400 + 800 = 1200$ At $(800, 0)$: $Z = 800 + 4(0) = 800$
The minimum value of $Z$ is $800$ at the point $(800, 0)$.
Final Answer: 800