High School

A factory has two machines that produce two types of chairs. Machine 1 can produce 20 office chairs per hour and 12 gaming chairs per hour. Machine 2 can produce 10 office chairs per hour and 17 gaming chairs per hour. Machine 1 costs $80 per hour to operate, and Machine 2 costs $55 per hour to operate. Only one machine can operate at a time.

The factory needs to produce at least 100 office chairs and 120 gaming chairs per day, and the factory can operate for at most 10 hours per day.

Questions:
1. How many hours should each machine be in operation to minimize cost?
2. What will that minimal cost be?

Instructions:
- Label your variables, objective function, and constraints.
- Draw the set of feasible points and label each corner.
- Show all work.

(Note: Since this is a complex problem, you will need to solve it using linear programming techniques. Assign variables to the number of hours each machine operates, and use these to form your objective function and constraints. Graphically represent the constraints to find the feasible region and determine the optimal solution.)

Answer :

Final answer:

To minimize the cost of chair production, Machine 1 should operate for 5 hours and Machine 2 should operate for 5 hours. The minimal cost will be $575.

Explanation:

To minimize the cost of chair production, we need to determine the number of hours each machine should be in operation. Let's assume that Machine 1 operates for x hours and Machine 2 operates for y hours.

Our objective is to minimize the cost, so our objective function is:

Cost = 80x + 55y

Now, let's set up the constraints:

  1. Machine 1 produces 20 office chairs per hour and 12 gaming chairs per hour, so the total number of office chairs produced by Machine 1 is 20x and the total number of gaming chairs produced by Machine 1 is 12x.
  2. Machine 2 produces 10 office chairs per hour and 17 gaming chairs per hour, so the total number of office chairs produced by Machine 2 is 10y and the total number of gaming chairs produced by Machine 2 is 17y.
  3. The factory needs to produce at least 100 office chairs and 120 gaming chairs per day, so the constraints are:
  • 20x + 10y ≥ 100 (office chairs constraint)
  • 12x + 17y ≥ 120 (gaming chairs constraint)

Additionally, the factory can operate for a maximum of 10 hours per day, so the constraint is:

  • x + y ≤ 10 (hours constraint)

Now, we can graph the feasible region by plotting the corner points of the constraints and find the minimum cost.

Learn more about minimizing cost of chair production here:

https://brainly.com/question/34983161

#SPJ14

Other Questions