A clothing company makes two basic products- shirts and pants. The company has 50 labor hours per day in the cutting department and 70 hours available in the sewing department. It takes 12 minutes to cut a shirt and 15 minutes to sew it. It takes 13 minutes to cut pants and 18 minutes to sew them. No more than 100 pairs of pants can be sold. The profit on a shirt is $5 and the profit on a pair of pants is $8. How many of each should the company produce in order to maximize profit?

Question

A clothing company makes two basic products- shirts and pants. The company has 50 labor hours per day in the cutting department and 70 hours available in the sewing department. It takes 12 minutes to cut a shirt and 15 minutes to sew it. It takes 13 minutes to cut pants and 18 minutes to sew them. No more than 100 pairs of pants can be sold. The profit on a shirt is $5 and the profit on a pair of pants is $8. How many of each should the company produce in order to maximize profit?

Answer 1

maximize p = 5s + 8t subject to

t <= 100
12s + 13t <= 3000
15s + 18t <= 4200

142 shirts and 100 trousers

Answer 2

To maximize profit, we need to determine how many shirts and pants should be produced. Let's start by defining our variables:

Let's assume that the number of shirts to be produced is "x" and the number of pants to be produced is "y".

Now, we can set up the constraints based on the given information:

1. Cutting Department Constraints:
- It takes 12 minutes to cut a shirt, so the total cutting time for shirts will be 12x.
- It takes 13 minutes to cut a pair of pants, so the total cutting time for pants will be 13y.
- The company has 50 labor hours per day available in the cutting department, which amounts to 3000 minutes (50 x 60). Therefore, the cutting time for shirts and pants should not exceed this limit: 12x + 13y ≤ 3000.

2. Sewing Department Constraints:
- It takes 15 minutes to sew a shirt, so the total sewing time for shirts will be 15x.
- It takes 18 minutes to sew a pair of pants, so the total sewing time for pants will be 18y.
- The company has 70 labor hours per day available in the sewing department, which amounts to 4200 minutes (70 x 60). Therefore, the sewing time for shirts and pants should not exceed this limit: 15x + 18y ≤ 4200.

3. Pants Quantity Constraint:
- The company can sell a maximum of 100 pairs of pants. Therefore, the number of pants produced, y, should not exceed 100: y ≤ 100.

4. Non-Negative Constraint:
- The number of shirts and pants produced cannot be negative: x ≥ 0 and y ≥ 0.

Now, we can formulate the objective function that represents the profit:

Profit = (profit per shirt) * (number of shirts) + (profit per pair of pants) * (number of pants)

Profit = 5x + 8y

To solve this problem, we need to graph the feasible region defined by these constraints and find the optimal solution.