Mathematics, 15.06.2020 12:57 cesar9895

Given the objective function C=3x−2y and constraints x≥0, y≥0, 2x+y≤10, 3x+2y≤18, identify the corner point at which the maximum value of C occurs.

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Answer from: eparikh7317

Given the objective function is C=3x-2y and the contraints as follows:

$x\geq 0\\ y\geq 0\\ 2x+y\leq 10\\ 3x+2y\leq 18$

Find the intersecting point of x=0

and

Substitute 0 for in 2x+y=10 .

$2(0)+y=10\\ y=10$

So, the intersecting point is (0,10) .

Find the intersecting point of y=0

and

Substitute 0 for in 2x+y=10 .

$2x+0=10\\ 2x=10\\ x=5$

So, the intersecting point is (5,0) .

Find the intersecting point of x=0

and

Substitute 0 for in 3x+2y=18 .

$3(0)+2y=18\\ 2y=18\\ y=9$

So, the intersecting point is (0,9) .

Find the intersecting point of y=0

and

Substitute 0 for in 3x+2y=18 .

$3x+2(0)=18\\ 3x=18\\ x=6$

So, the intersecting point is (6,0) .

Find the intesecting point of 2x+y=10,3x+2y=18

Add times 2x+y=10 to 3x+2y=18 .

$-2(2x+y)+3x+2y=-2(10)+18\\ -4x-2y+3x+2y=-20+18\\ -x=-2\\ x=2$

Substitute x=2 in 2x+y=10 :

$2(2)+y=10\\ 4+y=10\\y=6$

So, the intersecting point is (2,6) .

The origin (0,0)

is also a intersecting point of $x\geq 0,y\geq 0$ .

Corner points:

The corner points are the boundary points of the bounded region of the given constraints.

The bounded region of the given constraints is shown below.

From the graph notice that the shaded region is the required bounded region of the given constraints.The boundary points are (0,0),(5,0),(0,9),(2,6)

Evaluate the objective function C=3x-2y at these boundary points:

At (0,0) :

$C=3(0)-2(0)\\C=0$

At (5,0) :

$C=3(5)-2(0)\\C=15$

At (0,9) :

$C=3(0)-2(9)\\C=-18$

At (2,6) :

$C=3(2)-2(6)\\C=6-12\\C=-6$

From the above calculated values, one can notice that the maximum value of is 15 and it is obtained at (5,0) .

Hence, the maximum value of is 15 occurs at the corner point (5,0) .

Lear more about the maximizing problems here: link

Given the objective function C=3x−2y and constraints x≥0, y≥0, 2x+y≤10, 3x+2y≤18, identify the corne

Answer from: 182075

Step-by-step explanation:

Find the maximum value of

C = 3x -2y Objective function

subject to the following constraints.

Constraints

x ≥ 0

y ≥ 0

2x + y ≤ 10 vertex 1 : when x=0 then y=10 (0,10)

3x + 2y ≤ 18 vertex 2 : y=0, then x=6 ( 6,0)

two equations together to determine vertex 3 :

3x+2y = 18

2x+y = 10

x=2, y= 6

The feasible region determined by the constraints is

shown. The three vertices are (0, 10), and (6, 0), (0,9)

and (2,6)

First evaluate C = 3x -2 y at each of the vertices.

At (0, 10): C = 3(0) - 2(10) = -17

At (6, 0): C = 3(6) - 2(0) = 18

At ( 2,6) : C = 3(2) -2(6) = -6

At (0,9) : C = 3(0)-2(9)= -18

the maximum value occur on 18 when x=9 and y=0

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