Which one of the options given represents the feasible region of the linear programming model:

                                        Maximize 45X₁, + 60X₂

                                                           X₁ ≤ 45

                                                          X₂ ≤ 50

                                                          10X₁ + 10X₂ ≥ 600

                                                          25X₁, + 5X₂, ≤  750 

The correct option is Region Q.

To determine the feasible region of the linear programming model, we analyze each of the given constraints step-by-step:

1. Constraint 1:
$X_1\le 45$
This constraint is represented by a vertical boundary line at:
$X_1=45$
The feasible region must lie to the left of this vertical line.

2. Constraint 2:
$X_2\le 50$
This constraint is represented by a horizontal boundary line at:
$X_2=50$
The feasible region must lie below this horizontal line.

3. Constraint 3:
$10X_1+10X_2\ge 600$
Dividing both sides of the inequality by 10, we get the simplified constraint:
$X_1+X_2\ge 60$
The corresponding boundary line is $X_1+X_2=60$. Since the inequality is "greater than or equal to" (≥), the feasible area lies on or above this line (away from the origin).

4. Constraint 4:
$25X_1+5X_2\le 750$
Dividing both sides of the inequality by 5, we get the simplified constraint:
$5X_1+X_2\le 150$
The corresponding boundary line is $5X_1+X_2=150$. Since the inequality is "less than or equal to" (≤), the feasible area lies on or below this line (towards the origin).

Next, we find the intersection points of these boundaries to identify the corner points visible in the provided image:
- Intersection of the horizontal line $X_2=50$ and the line $X_1+X_2=60$:
$X_1+50=60\Rightarrow X_1=10$, yielding the corner point (10, 50).
- Intersection of the horizontal line $X_2=50$ and the line $5X_1+X_2=150$:
$5X_1+50=150\Rightarrow 5X_1=100\Rightarrow X_1=20$, yielding the corner point (20, 50).
- Intersection of the two lines $X_1+X_2=60$ and $5X_1+X_2=150$:
$(5X_1+X_2)-(X_1+X_2)=150-60\Rightarrow 4X_1=90\Rightarrow X_1=22.5$
Substituting back, we get:
$X_2=60-22.5=37.5$, yielding the corner point (22.5, 37.5).

By comparing the regions labeled in the diagram:
- Region P lies below both lines but is bounded by $X_1+X_2\le 60$ which violates Constraint 3.
- Region S lies to the right of $5X_1+X_2=150$ which violates Constraint 4.
- Region Q is the small shaded triangular region bounded by the vertices (10, 50), (20, 50), and (22.5, 37.5). It lies below $X_2=50$, above $X_1+X_2=60$, and below $5X_1+X_2=150$. This area perfectly satisfies all inequality constraints simultaneously.
Therefore, Region Q is the feasible region of the linear programming model.