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:
- 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.
- 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.
- 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.
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