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Find the optimal solution for the following problem. (Round your answers to 3 decimal places.)

Maximize C = 9x + 7y
subject to 8x + 10y ≤ 17
11x + 12y ≤ 25
and x ≥ 0, y ≥ 0.
1. what is the optimal value of x?

2. What is the optimal value of y?

3. What is the maximum value of the objective function?

Respuesta :

Answer:

Maximize C =[tex]9x + 7y[/tex]

[tex]8x + 10y \leq 17[/tex]

[tex]11x + 12y\leq 25[/tex]

and x ≥ 0, y ≥ 0

Plot the lines on graph

[tex]8x + 10y \leq 17[/tex]

[tex]11x + 12y\leq 25[/tex]

[tex]x\geq 0[/tex]

[tex]y\geq 0[/tex]

So, boundary points of feasible region are (0,1.7) , (2.125,0) and (0,0)

Substitute the points in  Maximize C

At  (0,1.7)

Maximize C =[tex]9(0) + 7(1.7)[/tex]

Maximize C =[tex]11.9[/tex]

At  (2.125,0)

Maximize C =[tex]9(2.125) + 7(0)[/tex]

Maximize C =[tex]19.125[/tex]

At   (0,0)

Maximize C =[tex]9(0) + 7(0)[/tex]

Maximize C =[tex]0[/tex]

So, Maximum value is attained at   (2.125,0)

So, the optimal value of x is 2.125

The optimal value of y is 0

The maximum value of the objective function is 19.125

Ver imagen wifilethbridge
MrRoyal

There are several ways to calculate optimal solution; one of them, is by using graphs.

  • The optimal value of x is 2.125
  • The optimal value of y is: 0
  • The maximum value of the objective function is: 19.125

The given parameters are:

[tex]\mathbf{Max\ C = 9x + 7y}[/tex]

[tex]\mathbf{8x + 10y \le 17}[/tex]

[tex]\mathbf{11x + 12y \le 25}[/tex]

[tex]\mathbf{x,y\ge 0}[/tex]

See attachment for the graphs of [tex]\mathbf{8x + 10y \le 17}[/tex] and [tex]\mathbf{11x + 12y \le 25}[/tex]

From the graph, the feasible regions are:

[tex]\mathbf{(x,y) = (0,1.7), (2.125,0)}[/tex]

Test these values in the objective function

[tex]\mathbf{Max\ C = 9x + 7y}[/tex]

[tex]\mathbf{C = 9 \times 0 + 7 \times 1.7 = 11.9}[/tex]

[tex]\mathbf{C = 9 \times 2.125 + 7 \times 0 = 19.125}[/tex]

So, the value of C is maximum at: [tex]\mathbf{(x,y) = (2.125,0)}[/tex]

So, we have:

  • The optimal value of x is 2.125
  • The optimal value of y is: 0
  • The maximum value of the objective function is: 19.125

Read more about maximizing functions at:

https://brainly.com/question/11212148

Ver imagen MrRoyal

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