Question

Given the system of constraints, name all vertices of the feasible region. Then find the maximum value of the given objective function.
Constraints: x is greater than or equal to 0
y is greater than or equal to 0
y is less than or equal to 1/3x+3
5 is greater than or equal to y+x

Objective Function: C=6x-4y

Answer 1

Answer with explanation:

The constraints are given by:

                $x\geq 0\\\\y\geq 0\\\\y\leq \dfrac{1}{3}x+3\\\\5\geq y+x$

• The first and second constraints tells that the solution must lie in the first quadrant including the origin )

( Since, in the first quadrant both the x and y values are positive )

• Also, the third inequality is a solid line ( since the inequality is not strict i.e. it is a inequality with a equality sign ) that passes through (-9,0) and (0,3) and the shaded region is towards the origin ( since the inequality passes the zero point test ).

• The fourth inequality is a solid line ( since the inequality is strict ) that passes through (5,0) and (0,5) and the shaded region is towards the origin ( since the inequality passes the zero point test)

Ques 1)

The feasible region that is formed with the help of these inequality is:

                     Quadrilateral ABCD.

Also, the vertices of the feasible region are:

A(0,3)

B(1.5,3.5)

C(5,0)

D(0,0)

Ques 2)

We know that the objective function is checked at the boundary points of the feasible region i.e. vertices of the feasible region.

The objective function is:

C=6x-4y

• We check at A(0,3)

              C=6×0-4×3

                 i.e.

               C= -12

• at B(1.5,3.5)

            C=6×1.5-4×3.5

                  i.e.

                C= -5

• at C(5,0)

           The value of objective function is:

                 C=6×5-4×0

                        i.e.

                      C=30

• and at D(0,0)

                      C=6×0-4×0

                          i.e.

                        C=0

The maximum value of the objective function is: 30

and it is obtained at the vertex C(5,0)

Answer 2

Attached the solution and work.