Question

Graph the system of constraints and find the value of x and y that maximize the objective function.

Constraints
x≥0
y≥0
y≤1/5+2
5≥y+x
Objective Function:
C=7x-3y
Answers:
(0,0)
(2,3)
(5,0)
(0,3)

Answer 1

Given objective function
`C= 7x-3y`

The constraints are:

1.
`x\geq 0`

2.
`y\geq 0`

3.
`y\leq (x)/(5) +2`

4.
`5\geq y+x`

The attachment attached shows the feasible region between the given constraints.

The end points of the feasible region are (0,0) ( 0,2) (2.5,2.5) (5,0)

For (0,0), the objective function is
`7(0)-3(0) = 0`

For (0,2) , objective function is
`7(0)-3(2) = -6`

For (2.5,2.5) the objective function is
`7(2.5)-3(2.5)= 10`

For (5,0), the objective function is
`7(5)-3(0) = 35`

The objective function is highest 35 for the point (5,0).

Option C is the correct answer.

Answer 2

From the graph, we can see the coordinates of the vertices of feasible region (common region of the shaded portion) are (0,2) (2.5, 2.5) and (5,0).

We are given

Objective Function:C=7x-3y

Plugging each of the coordinate (0,2) (2.5, 2.5) and (5,0) on by on in

C=7x-3y function, we get

For (0,2)

C=7(0) -3(2) = 0-6 = -6.

For (2.5, 2.5)

C = 7(2.5) -3(2.5) = 17.5 -7.5 = 10.0.

For (5,0)

C = 7(5) -3(0)= 35 -0 = 35.

Because in the given options (0,0)(2,3)(5,0)(0,3), only (5,0) is the only coordinate of the vertex of feasible region.

Also we can see that (5,0) shows maximize the objective function.

Therefore, correct option is (5,0).