Are you struggling with linear system modeling assignments? Don't worry; you're not alone. Many students find this topic challenging due to its complex nature and mathematical intricacies. However, with the right guidance and understanding, you can conquer these assignments with ease. In this blog post, we'll walk you through a tough linear system modeling assignment question and provide a detailed step-by-step guide on how to approach it.

Assignment Question: Consider a scenario where a company is manufacturing two types of products, Product A and Product B. The company produces 200 units of Product A and 150 units of Product B per day. Each unit of Product A requires 3 hours of labor and 2 hours of machine time, while each unit of Product B requires 2 hours of labor and 4 hours of machine time. The company has a total of 800 hours of labor and 600 hours of machine time available per day. Determine the maximum profit the company can achieve if the profit from each unit of Product A is $50 and from each unit of Product B is $60.

Step-by-Step Guide:

1. Identify Variables: First, let's identify the variables in our problem:

   • $x$: Number of units of Product A produced per day
   • $y$: Number of units of Product B produced per day

2. Formulate Constraints: We need to ensure that the company does not exceed the available labor and machine time. The constraints are:

   • Labor Constraint: 3x+2y≤800 (Total labor hours available per day)
   • Machine Constraint: $2x+4y\le 600$ (Total machine hours available per day)

3. Define Objective Function: The objective is to maximize profit. The profit function is:

   • $P=50x+60y$

4. Graphical Representation: Plot the constraints on a graph and shade the feasible region.

5. Optimization: Identify the corner points of the feasible region and evaluate the objective function $P$ at each point.

6. Find Maximum Profit: Determine which corner point yields the maximum profit.

Sample Solution:

Let's solve the problem step-by-step:

1. Identify Variables:

   • $x$: Number of units of Product A
   • $y$: Number of units of Product B

2. Formulate Constraints:

   • $3x+2y\le 800$
   • $2x+4y\le 600$

3. Define Objective Function:

   • $P=50x+60y$

4. Graphical Representation:

   • Plot the lines corresponding to each constraint and shade the feasible region.

5. Optimization:

   • Find the corner points of the feasible region: $(0,0),(0,150),(200,0),(225,75)$.
   • Evaluate the objective function $P$ at each corner point:
     • For $(0,0)$, $P=0$.
     • For $(0,150)$, $P=60\times 150=9000$.
     • For $(200,0)$, $P=50\times 200=10000$.
     • For $(225,75)$, $P=50\times 225+60\times 75=20250$.

6. Find Maximum Profit:

   • The maximum profit occurs at the point $(225,75)$ with a profit of $20,250.

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