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Chapter 7: Problem 66

\(\mathrm{A}\) store sells two models of laptop computers. Because of the demand, the store stocks at least twice as many units of model \(\mathrm{A}\) as of model \(\mathrm{B}\). The costs to the store for the two models are \(\$ 800\) and \(\$ 1200\), respectively. The management does not want more than \(\$ 20,000\) in computer inventory at any one time, and it wants at least four model A laptop computers and two model B laptop computers in inventory at all times. Find and graph a system of inequalities describing all possible inventory levels.

Short Answer

Expert verified
The system of inequalities representing possible inventory levels is \(x \geq 2y\), \(800x + 1200y \leq 20000\), \(x \geq 4\) and \(y \geq 2\). The graph representing these inequalities gives us a shaded region describing the problem constraints. This region contains all the possible quantities of model A and B laptops that the store can stock.

Step by step solution

01

Define the Variables

Let \(x\) represent the amount of model A laptops and \(y\) represent the amount of model B laptops. These represent quantities stocked in the store.
02

Inequality for the first information

The store stocks at least twice as many units of model A as of model B. So we can write this information as an inequality \(x \geq 2y\). This inequality represents that the number of Model A laptops \(x\) is greater or equal to twice the number of model B laptops \(y\).
03

Inequality for the second information

The management does not want more than $20,000 in computer inventory at any one time. Considering the costs for models A and B are \(\$ 800\) and \(\$ 1200\) respectively, we reach the inequality \(800x+1200y \leq 20000\). This inequality represents that the total cost of the computers cannot exceed \$20000.
04

Inequality for the third information

The store wants at least four model A laptops and two model B laptops. Thus, we have two more inequalities: \(x \geq 4\) and \(y \geq 2\). These inequalities ensure the store keeps at least 4 of model A and 2 of model B at all times.
05

Graph the system of inequalities

Once the system of inequalities is set up, we need to graph these inequalities on a Cartesian plane. The solution will be the shaded overlap of all inequalities.The horizontal axis is for Model A laptops \(x\), with values starting from 4. The vertical axis is for the Model B laptops \(y\), with values starting from 2. The line \(x = 2y\) marks the boundary of the region where the store stocks at least twice as many units of model A as of model B. With the inequality \(800x + 1200y \leq 20000\), we have another bounding line. The shaded solution region describes all possible inventory levels fulfilling the management's conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Inequalities
Linear inequalities are mathematical expressions that involve a linear function and an inequality sign. In the context of this problem, they help us determine feasible solutions for inventory levels.

Consider the inequality:
  • \(x \geq 2y\) - This means the number of Model A laptops must be at least twice the number of Model B laptops.
  • \(800x + 1200y \leq 20000\) - Here, the total cost of all laptops in stock must not exceed $20,000.
  • \(x \geq 4\) and \(y \geq 2\) - These ensure minimum stock levels for both models.
Each inequality creates a boundary on a graph, helping visualize where solutions are possible. Adjusting these inequalities allows stores to meet their constraints while maximizing efficiency.
Inventory Management
Inventory management involves optimizing stock levels to meet customer demands while minimizing costs. In this exercise, the store is managing the number of laptops it keeps in stock.

Key aspects include:
  • Ensuring enough stock to meet demand without overstocking. The inequalities \(x \geq 2y\) and \(800x + 1200y \leq 20000\) help achieve this balance.
  • Maintaining minimum required levels for models A and B as expressed by \(x \geq 4\) and \(y \geq 2\).
These constraints ensure the store operates efficiently, meeting customer needs and keeping costs within a feasible range.
Graphical Representation
Graphical representation is a method used to visualize solutions of inequalities and systems of inequalities. By plotting these on a Cartesian plane, we can easily see feasible solutions for the problem.

Here's how we graph each inequality:
  • The line \(x = 2y\) marks one boundary. The shaded area above this line indicates where \(x\) is at least twice \(y\).
  • The inequality \(800x + 1200y \leq 20000\) is another boundary, represented by a line in the graph. Shading below this line demonstrates where the inventory cost stays within budget.
  • Additional lines for \(x \geq 4\) and \(y \geq 2\) ensure minimum inventory levels.
The intersection of these shaded areas gives us the feasible region, showing all possible inventory levels that meet the store's conditions.
Constraint Optimization
Constraint optimization is about finding the best possible solution within given constraints. In inventory management, it means stocking the right amount of products without exceeding budget constraints.

In this exercise:
  • We use \(x \geq 2y\) to ensure demand for Model A meets market needs while \(800x + 1200y \leq 20000\) ensures financial constraints are met.
  • \(x \geq 4\) and \(y \geq 2\) constraints ensure basic availability for both models.
The goal is to find the optimal point within the feasible region on the graph, where all constraints are satisfied. This approach enables the store to operate efficiently, keeping both supply and demand in balance.

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