OKay My question is:
Two factories manufacture 3 different grades per paper. the company that owns the factories has contracts to supply at least 16 tons of low grade, 5 tons of medium grade, and at least 20 tons of high grade paper. It cost $1000 per day to operate the first factory and 2000 per day to operate the second. Factory 1 produces 8 tons of low grade, 1 ton of medium grade, and 2 tons of high grade paper in day's operation. Factory 2 produces 2 tons of low grade, 1 ton of medium grade and 7 tons of high grade per day. how many days should each factory be in operation in order to fill the most economically?
Also i have to assign variables to x and y and state a goal plus represent the goal with an equation
then i have to write inequalities that represent the restrictions of this problem in a chart. Then i have to grade the system of inequalities, shade the solution region, and use corner points to analyze data and find the solution. Finally I write my solution in a complete sentence.
Help?
So the instructions seem to be clear. Do you have a problem, or where is the problem if there is any?
It's just I'm stuck at picking out what i need and such.. Not very good with word problems..
You can take the question and split it up into constraints and costs.
Requirements:
"supply at least 16 tons of low grade, 5 tons of medium grade, and at least 20 tons of high grade paper."
So minimum quantities = (16,5,20)
Supplied by Factory 1, F1 = (8,1,2)/$1000
Supplied by F2 = (2,1,7)/$2000
So the constraints are:
8F1+2F2 ≥ 16
F1+F2 ≥ 5
2F1+7F2 ≥ 20
F1 ≥ 0
F2 ≥ 0
Cost=1000F1+2000F2
Your graph will have F1 in the x-axis, F2 in the y-axis. Plot the constraint equations, and shade the solution (feasible) region. Find the optimal solution.
To solve this problem, let's first assign variables to represent the number of days each factory should be in operation. Let's use x for Factory 1 and y for Factory 2.
Now, let's state the goal: we want to minimize the cost of operations while satisfying the supply requirements.
To represent the goal with an equation, we need to consider the cost of operations for each factory. The cost for Factory 1 operating for x days is 1000x, and the cost for Factory 2 operating for y days is 2000y. So, our goal equation is:
Total Cost = 1000x + 2000y
Next, let's write the inequalities that represent the restrictions of the problem. We have the following supply requirements:
Low grade paper: at least 16 tons
Medium grade paper: 5 tons
High grade paper: at least 20 tons
The amounts produced by each factory per day are:
Factory 1: 8 tons low grade, 1 ton medium grade, 2 tons high grade
Factory 2: 2 tons low grade, 1 ton medium grade, 7 tons high grade
Based on these, the supply restrictions can be written as inequalities:
Low grade paper: 8x + 2y >= 16
Medium grade paper: 1x + 1y >= 5
High grade paper: 2x + 7y >= 20
Now, let's graph these inequalities to visualize the solution region.
Finally, to find the optimal solution, we need to look at the corner points of the solution region and evaluate the objective function (total cost) at these points. The point that minimizes the total cost will be the answer to our problem.
You can solve the system of inequalities using a graphical method or solve it using linear programming techniques. Once the solution region is shaded, you can find the corner points by looking at the intersection points of the lines that form the region.
Once you have the corner points, substitute the values into the goal equation (Total Cost) to find the lowest cost solution. The values of x and y that yield the minimum cost will tell you how many days each factory should be in operation.
Finally, you can write your solution in a complete sentence, stating the number of days each factory should be operated to fill the order with the lowest cost.
I hope this helps you in solving the problem!