Question 1176641: A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $2.00 and the profit on a jacket is $1.50. Both pants and jackets require the work of sewing operators and cutters. There are 60 minutes of sewing operator time and 48 minutes of cutter time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket.
Find the number of pants and jackets the manufacturer should make in order to maximize the profit.
? pairs of pants
? jackets
What is the maximum profit? $
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this linear programming problem step-by-step.
**1. Define Variables**
* Let 'x' be the number of ski pants.
* Let 'y' be the number of ski jackets.
**2. Formulate the Objective Function**
The objective is to maximize profit. The profit function (P) is:
* P = 2x + 1.5y
**3. Formulate the Constraints**
* **Sewing Time Constraint:** 8x + 4y ≤ 60
* **Cutter Time Constraint:** 4x + 8y ≤ 48
* **Non-negativity Constraints:** x ≥ 0, y ≥ 0
**4. Simplify the Constraints**
* **Sewing Time:** 2x + y ≤ 15
* **Cutter Time:** x + 2y ≤ 12
**5. Find the Corner Points**
* **Point 1 (Origin):** (0, 0)
* **Point 2 (x-intercept of sewing time):** Set y = 0 in 2x + y = 15. 2x = 15, x = 7.5. (7.5, 0)
* **Point 3 (y-intercept of cutter time):** Set x = 0 in x + 2y = 12. 2y = 12, y = 6. (0, 6)
* **Point 4 (Intersection of the two constraints):**
Solve the system of equations:
* 2x + y = 15
* x + 2y = 12
Multiply the second equation by 2:
* 2x + 4y = 24
Subtract the first equation from this:
* 3y = 9
* y = 3
Substitute y = 3 into x + 2y = 12:
* x + 6 = 12
* x = 6
Intersection point: (6, 3)
**6. Evaluate the Objective Function at Each Corner Point**
* **(0, 0):** P = 2(0) + 1.5(0) = 0
* **(7.5, 0):** P = 2(7.5) + 1.5(0) = 15
* **(0, 6):** P = 2(0) + 1.5(6) = 9
* **(6, 3):** P = 2(6) + 1.5(3) = 12 + 4.5 = 16.5
**7. Determine the Maximum Profit**
The maximum profit is $16.50, which occurs at the point (6, 3).
**Answers**
* **Pairs of pants:** 6
* **Jackets:** 3
* **Maximum profit:** $16.50
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