Linear Programming Problem

The given constraints are:

1.

2.

We need to check which of the points satisfy both constraints.

Checking for each point:

For :

1. , which does not satisfy .

2. , which satisfies .

This point is not in the feasible region.

For :

1. , which satisfies .

2. , which does not satisfy .

This point is not in the feasible region.

For :

1. , which does not satisfy .

2. , which does not satisfy .

This point is not in the feasible region.

For :

1. , which does not satisfy .

2. , which does not satisfy .

This point is not in the feasible region.

For :

1. , which satisfies .

2. , which satisfies .

This point satisfies both constraints and is in the feasible region.

The correct answer is (1, 3).

To find the maximum value of , we need to evaluate the objective function at the vertices of the feasible region. From the graph, the coordinates of the vertices are:
A(20, 0)
B(10, 50)
C(0, 60)
We calculate at each of these points: For point A(20, 0): For point B(10, 50): For point C(0, 60): The maximum value of occurs at point B(10, 50) with .

We first graph the constraints to identify the feasible region. The equations for the constraints are:
1.
2.
3.
Now, we find the points of intersection for the corner points of the feasible region.
The first constraint intersects the x-axis at , giving the point .
The second constraint intersects the x-axis at , giving the point .
The first and second constraints intersect each other by solving the system of equations: Subtract the second equation from the first: Substitute into : So, the point of intersection is . Thus, the coordinates of the corner points are .

The correct option is (A) :

In a linear programming problem, the restrictions under which the objective function is to be optimized are called constraints.

The key components are:

• Decision variables (A): The variables that represent quantities to be determined
• Objective function (B): The function to be maximized or minimized
• Constraints (C): The restrictions or limitations on the decision variables

Since the question specifically asks about the restrictions, the correct answer is (C) constraints.