Math Problem Statement
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Solution
This problem involves optimizing the production of two types of toy airplane models (A and B) based on constraints of assembly and packaging time, while maximizing the manufacturer's profit. Let's break it down step by step:
Variables:
Let:
- = number of units of Model A to be produced
- = number of units of Model B to be produced
Constraints:
-
Assembly time:
- Each unit of Model A takes 20 minutes to assemble.
- Each unit of Model B takes 25 minutes to assemble.
- The total available assembly time is 3000 minutes.
-
Packaging time:
- Each unit of Model A takes 10 minutes to package.
- Each unit of Model B takes 5 minutes to package.
- The total available packaging time is 1200 minutes.
Objective:
- The goal is to maximize profit.
- Profit for Model A: $9 per unit.
- Profit for Model B: $12 per unit. The objective function to maximize is:
Non-Negativity constraints:
Linear Programming Problem:
We now have the system of inequalities:
We need to solve this system graphically or using another method (e.g., simplex method) to find the values of and that maximize .
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Algebra
Formulas
20x + 25y ≤ 3000
10x + 5y ≤ 1200
Profit: Z = 9x + 12y
Theorems
Linear Inequalities
Simplex Method
Suitable Grade Level
Grades 10-12
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