a paint supplier has two machines that prouce both indoor paint and outdoor paint. to meet its contractual obligations,the company must produce at least 60 gallons of indoor paint and 100 gallons of outdoor paint. machine 1 makes 3 gallons of indoor paint and 10 gallons of outdoor paint per hour.machine 2 makes 4 gallons of indoor paint and 5 gallons of outdoor paint per hour. it costs $28 per hour to run machine 1 and $33 per hour t run machine 2.
how many hours should each machine be operated to fulfill the contract at minimum cost?
To understand how many hours each machine should be operated to fulfill the contract at minimum cost, we can set up a linear programming problem.
Let's define the variables:
Let x1 be the number of hours machine 1 is operated.
Let x2 be the number of hours machine 2 is operated.
We can then formulate the objective function to be minimized:
Minimize the cost, which is 28x1 (machine 1 cost) + 33x2 (machine 2 cost).
Subject to the following constraints:
1. Machine 1 produces 3 gallons of indoor paint per hour, and the company must produce at least 60 gallons. So, 3x1 ≥ 60.
2. Machine 1 produces 10 gallons of outdoor paint per hour, and the company must produce at least 100 gallons. So, 10x1 ≥ 100.
3. Machine 2 produces 4 gallons of indoor paint per hour, and the company must produce at least 60 gallons. So, 4x2 ≥ 60.
4. Machine 2 produces 5 gallons of outdoor paint per hour, and the company must produce at least 100 gallons. So, 5x2 ≥ 100.
5. The number of hours for both machines must be non-negative. x1 ≥ 0 and x2 ≥ 0.
Let's solve this linear programming problem to find the optimal values for x1 and x2.
To do this, we can use optimization software (such as Excel Solver or Python libraries like PuLP or scipy.optimize) or a graphical method (by graphing the feasible region and finding the optimal solution).
By solving this problem, we will find the values for x1 and x2 that minimize the cost while fulfilling the contractual obligations.