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Sporting Equipment, Inc. makes two types of balls: Soccer balls and Cork balls. The making of each soccer ball and cork ball requires 3 hours and 4 hours of production time, respectively. A total of 500 production hours are available during the next month At least 150 balls, combined, must be produced. The production cost for each Soccer ball is $9 and each Cork ball is $7.Formulate and solve a linear programming model minimizing production costs.

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funmilaciousfunmie78

Answer:

100 soccer balls and 50 cork balls must be produced

Step-by-step explanation:

Let the number of soccer ball = p  and number of cork ball = q

Production hours of p = 3 hours

Production hours of q = 4 hours

Then the production time equation would be:

3p + 4q ≤500

The number of the balls must be at least 150

p + q ≥150

x ≥ 0     y ≥ 0

Minimum production r = 9p + 7q

Therefore 100 soccer balls and 50 cork balls must be produced

Ver imagen funmilaciousfunmie78
MrRoyal

The linear programming model is:

  • Minimize [tex]\mathbf{Z = 9x + 7y}[/tex]
  • Subject to [tex]\mathbf{3x + 4y \le500}[/tex], [tex]\mathbf{x + y \ge 150}[/tex] , [tex]\mathbf{x,y \ge 0}[/tex]
  • Sporting Equipment, Inc should produce at least 100 soccer balls and at most 50 co rk balls.

Represent soccer balls with x and co rk balls with y.

The given parameters are:

Production hours:

[tex]\mathbf{x=3\ hours}[/tex]

[tex]\mathbf{y=4\ hours}[/tex]

[tex]\mathbf{Available = 500\ hours}[/tex]

So, we have:

[tex]\mathbf{3x + 4y \le500}[/tex]

Number of balls

[tex]\mathbf{Minimum = 150}[/tex]

So, we have:

[tex]\mathbf{x + y \ge 150}[/tex]

The cost is given as: $9 per soccer ball, and $7 per co rk ball.

So, the objective function is:

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

Hence, the linear programming model is:

Minimize [tex]\mathbf{Z = 90x + 75y}[/tex]

Subject to

[tex]\mathbf{3x + 4y \le500}[/tex]

[tex]\mathbf{x + y \ge 150}[/tex]

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

Express [tex]\mathbf{x + y \ge 150}[/tex] as an equation

[tex]\mathbf{x = 150 - y}[/tex]

Substitute [tex]\mathbf{x = 150 - y}[/tex] in [tex]\mathbf{3x + 4y \le500}[/tex]

[tex]\mathbf{3(150 - y) + 4y \le500}[/tex]

[tex]\mathbf{450 - 3y + 4y \le500}[/tex]

Subtract 450 from both sides

[tex]\mathbf{y \le50}[/tex]

Substitute 50 for y in [tex]\mathbf{x + y \ge 150}[/tex]

[tex]\mathbf{x + 50 \ge 150}[/tex]

Subtract 50 from both sides

[tex]\mathbf{x \ge 100}[/tex]

Hence, Sporting Equipment, Inc should produce at least 100 soccer balls and at most 50 co rk balls.

Read more about linear programming model at:

https://brainly.com/question/13720072

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