Problem 66 - Precalculus with Limits

Question

Inventory \(A\) store sells two models of laptop computers. Because of the demand, the store stocks at least twice as many units of model \(A\) as of model \(B\) . 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.

Step-by-Step Solution

Answer

The system of inequalities as per the given requirements is \(x \geq 2y\), \(x \geq 4\), \(y \geq 2\), and \(800x + 1200y \leq 20000\). The possible inventory levels will be the combined area satisfying these inequalities when graphed.

Step 1: Identify Variables

Let the number of model A laptops be represented as 'x', while the number of model B laptops be represented as 'y'.

Step 2: Formulate Inequalities from Requirements

From the conditions stated in the problem: 1. The store stocks at least twice as many units of model A as of model B. Therefore, \(x \geq 2y\).2. There must be at least four model A laptop computers and two model B laptop computers in inventory. Therefore, \(x \geq 4\) and \(y \geq 2\).3. The store does not want more than $20,000 in computer inventory at any time and the costs for model A and B are $800 and $1200, respectively. Therefore, \(800x + 1200y \leq 20000\).

Step 3: Graph the inequalities

Plot these inequalities on a coordinate plane, using x as the horizontal axis (for Model A) and y as the vertical axis (for Model B). The required area will be where all inequality conditions are satisfied simultaneously.

Key Concepts

Linear Inequalities

Linear inequalities allow us to make comparisons between linear expressions. They are like linear equations, but instead of an "equals" sign, they use symbols like "greater than" (\(>\)), "less than" (\(<\)), "greater than or equal to" (\(\geq\)), and "less than or equal to" (\(\leq\)). In the example at hand, we are using inequalities to express relationships and constraints around inventory levels relative to computer demand and financial limitations.

A linear inequality in two variables, such as \(x\) and \(y\), appears in the form \(ax + by \leq c\), where \(a\), \(b\), and \(c\) are constants. The solution set for a linear inequality is always a region on the graph, which helps to visualize, understand, and solve real-world problems, like inventory management in a store setting.

Graphing Inequalities

Graphing inequalities involves plotting the solutions of these inequalities on a coordinate plane. Let's focus on the inequalities derived from the example:

• \(x \geq 2y\)
• \(x \geq 4\)
• \(y \geq 2\)
• \(800x + 1200y \leq 20000\)

The first step is to draw the boundary lines for each inequality. For example, for \(x \geq 2y\), plot the line \(x = 2y\).

Remember:

• Use solid lines if the inequality includes equals (\(\leq\) or \(\geq\)), or dashed lines if it doesn't (\(<\) or \(>\)).

Once the lines are drawn, shade the area that satisfies the inequality. For \(x \geq 2y\), shade above the line, meaning for every point in this region, \(x\) will always be more than twice \(y\). The same process applies for other inequalities. The overlapping shaded region represents the solution set where all conditions are met together.

Inventory Optimization

Inventory optimization involves finding the best balance between different restrictions to ensure efficiency and cost-effectiveness in inventory management. In the context of the store selling laptops, the challenge is to find a feasible combination of model A and model B laptops that satisfies demand constraints while not exceeding the financial budget of \(\$20000\).

This process involves evaluating the system of inequalities. The inequalities guide decisions to ensure that the layout of laptops maximizes inventory space and budget, while meeting demand requirements. Effective inventory optimization leads to better resource allocation and increased satisfaction of customer needs without excess expenditure.

Laptop Inventory Management

Laptop inventory management is about maintaining the right number of laptops in stock to meet customer demand while keeping costs manageable. With various model preferences and budgetary limitations, it becomes crucial to manage different variables carefully.

Key elements include:

• Number of units of each laptop model in stock.
• Costs associated with each model.
• Minimum and maximum stock levels to maintain efficiency.

Managing inventory through a system of inequalities allows a store or business to determine how many laptops of each kind should be ordered and kept on hand. This ensures that they can cater to customer demand and prevent overstocking, which ties up capital. Such systems help retailers strike a balance between pre-empting increased sales and managing storage costs.