24. Impact Printing makes two kinds of computer paper using premium or ordinary quality stock. They have a contract to supply at least 5000 cases of paper. There is only enough stock to make 4000 cases of premium paper, but ample stock for ordinary paper. Both kinds are made with the same machine and 1200 hours of machine time are available. Premium paper takes 18 minutes per case to make and ordinary paper takes 12 minutes per case. The profit on each is $4/case and $3/case, respectively.

ware is the answer

If there are p cases of premium, and s cases of ordinary stock, then what do you know?

p+s >= 5000
p <= 4000
18p + 12s <= 1200
profit = 4p+3s

Now, when you decide what it is you want to know, you can use a little linear programming to get your answer.

To determine the optimal production quantities of premium and ordinary paper, we can set up a linear programming problem.

Let's define the decision variables:
x = number of cases of premium paper to produce
y = number of cases of ordinary paper to produce

Objective function:
The objective is to maximize the total profit, which can be represented as:
Profit = 4x + 3y

Constraints:
1. The total number of cases produced cannot exceed 5000:
x + y ≤ 5000

2. The number of cases of premium paper cannot exceed the available stock of 4000:
x ≤ 4000

3. The total machine time used cannot exceed 1200 minutes:
18x + 12y ≤ 1200

4. The number of cases produced cannot be negative:
x, y ≥ 0

Now, let's solve the linear programming problem step-by-step.

Step 1: Set up the objective function:
Maximize Profit = 4x + 3y

Step 2: Set up the constraints:
x + y ≤ 5000
x ≤ 4000
18x + 12y ≤ 1200
x, y ≥ 0

Step 3: Graph the feasible region:
Graph the inequalities on a coordinate plane and shade the feasible region where all constraints are satisfied.

Step 4: Identify the corner points:
Find the coordinates of the vertices (corner points) of the feasible region.

Step 5: Evaluate the objective function at each corner point:
Calculate the profit at each corner point by substituting the coordinates into the objective function.

Step 6: Determine the maximum profit:
Identify the corner point(s) that yield the maximum profit.

Step 7: Calculate the optimal production quantities:
Use the identified corner point(s) to determine the optimal production quantities.

By following these steps, you should be able to determine the optimal production quantities of premium and ordinary paper to maximize the profit while meeting all the given constraints.

To determine the optimal production plan for Impact Printing in order to maximize their profit, we need to consider the constraints and objectives of the problem.

1. Let's define the variables:
Let's say x represents the number of premium paper cases to be produced.
Let's say y represents the number of ordinary paper cases to be produced.

2. Establish the objective function:
The objective is to maximize the profit.
The profit for premium paper cases is $4 per case, and the profit for ordinary paper cases is $3 per case.
So the objective function can be written as:
Profit = 4x + 3y

3. Define the constraints:
- The contract requires the supply of at least 5000 cases of paper.
So the constraint is: x + y ≥ 5000

- There is only enough stock to make 4000 cases of premium paper.
So the constraint is: x ≤ 4000

- Both kinds are made with the same machine, and 1200 hours of machine time are available.
Premium paper takes 18 minutes per case, and ordinary paper takes 12 minutes per case.
So the machine time constraint can be written as:
18x + 12y ≤ 1200

4. Solve the linear programming problem:
We have an objective function to maximize and three constraints.

Use a graphical method or linear programming software to solve the problem and find the optimal values of x and y that maximize the profit.

The optimal solution will provide the production quantities of premium and ordinary paper cases that maximize the profit while satisfying all the given constraints.