Question

The maximum value of the function $z=4x+3y$ subject to the constraints $3x+2y\ge 160,5x+2y\ge 200$ , $x+2y\ge 80,x\ge 0,y\ge 0$ is

• A. $320$
• B. $300$
• C. $250$
• D. no maximum

Answer 1

The correct option is D no maximum

First, plot the lines given and then shade the region that satisfies all the given inequalities. This is the feasible region. It can be seen that the feasible region is unbounded. (Figure- 1)
By solving the equations of the lines, the corner points can be seen to be: $(0,100),(20,50),(40,20),(80,0)$
According to the Theorem of Linear Programming, if a maximum exists, it must be at one of the corner points.
Evaluating $z=4x+3y$ at the corner points gives:
$(x,y)$ $z$
$(0,100)$ $300$
$(20,50)$ $230$
$(40,20)$ $220$
$(80,0)$ $320$ $\leftarrow$ Maximum
As the feasible region is unbounded, we must check whether the maximum obtained above is in fact a maximum. For that we must plot $4x+3y>320$
As plotted in Figure-2, it can be seen that this does have common points with the feasible region. This means that there is $\text{no maximum.}$