Find graphically, the maximum value of $Z=2x+5y$, subject to constraints given below: $2x+4y≤8,3x+y≤6,x+y≤4,x≥0,y≥0$

10

The correct answer is Option (4) → 10

Draw the lines $x+2y=4$ (passes through (4, 0), (0, 2));

$3x+y=6$ (passes through (2, 0), (0, 6))

$x+y=4$ (passes through (4, 0), (0, 4))

Shade the region satisfied by the given inequalities.

The shaded region in the adjoining figure gives the feasible region determined by the given inequalities. Solving $3x+y=6$ and $x+2y=4$ simultaneously, we get

$x=\frac{8}{5}$ and $y=\frac{6}{5}$

We observe that the feasible region OABC is a convex polygon and bounded and has corner points $O(0, 0), A(2, 0), B(\frac{8}{5}, \frac{6}{5})$ and $C(0,2)$.

The optimal Solution occurs at one of the corner points

At $O(0,0),Z=2.0+5.0=0$;

at $A(2,0),Z=2.2+5.0=4$;

at $B(\frac{8}{5},\frac{6}{5}),Z=2.\frac{8}{5}+5.\frac{6}{5}=\frac{46}{5}$;

at $C(0,2),Z=2.0+5.2=10$.

Therefore, Z has maximum value at C and maximum value = 10.