The maximum value of the object function Z = 5x + 10 y subject to the constraints x + 2y ≤ 120, x + y ≥ 60, x – 2y ≥ 0, x ≥ 0, y ≥ 0 is

The correct answer is 600.

To find the maximum value of the objective function:

$Z=5x+10y$

we need to determine the feasible region defined by the following system of linear inequalities (constraints):

1. $x+2y\le 120$

2. $x+y\ge 60$

3. $x-2y\ge 0$

4. $x\ge 0,y\ge 0$

Step 1: Identify the Boundary Lines
We first convert the inequalities into equations to plot the boundary lines:
• Line 1: $x+2y=120$
• Line 2: $x+y=60$
• Line 3: $x-2y=0$ (or $x=2y$)

Step 2: Find the Corner Points (Vertices) of the Feasible Region
The feasible region is bounded by the intersections of these lines under the given constraints:

• Intersection of Line 1 ($x+2y=120$) and Line 3 ($x=2y$):
Substituting $x=2y$ into Line 1:
$2y+2y=120\Rightarrow 4y=120\Rightarrow y=30$
Thus, $x=2\left(30\right)=60$. This gives the corner point A(60, 30).

• Intersection of Line 2 ($x+y=60$) and Line 3 ($x=2y$):
Substituting $x=2y$ into Line 2:
$2y+y=60\Rightarrow 3y=60\Rightarrow y=20$
Thus, $x=2\left(20\right)=40$. This gives the corner point B(40, 20).

• Boundary points on the x-axis ($y=0$):
For Line 2 with $y=0$, we have $x=60$, giving the corner point C(60, 0).
For Line 1 with $y=0$, we have $x=120$, giving the corner point D(120, 0).

All four points satisfy all the given inequality constraints, forming the vertices of the feasible region.

Step 3: Evaluate the Objective Function Z at Each Corner Point
We substitute the coordinates of each vertex into $Z=5x+10y$:

• At point A(60, 30):
$Z=5\left(60\right)+10\left(30\right)=300+300=600$

• At point B(40, 20):
$Z=5\left(40\right)+10\left(20\right)=200+200=400$

• At point C(60, 0):
$Z=5\left(60\right)+10\left(0\right)=300$

• At point D(120, 0):
$Z=5\left(120\right)+10\left(0\right)=600$

Comparing the values, the maximum value of the objective function $Z$ is 600 (which is attained at the vertices A and D, and consequently at all points along the line segment connecting them).