Question

Kayak Inventory A store sells two models of kayaks. Because of the demand, it is necessary to stock at least twice as many units of model \(\mathrm{A}\) as units of model \(\mathrm{B}\). The costs to the store for the two models are \(\$ 500\) and \(\$ 700\), respectively. The management does not want more than \(\$ 30,000\) in kayak inventory at any one time, and it wants at least six model A kayaks and three model B kayaks in inventory at all times. (a) Find a system of inequalities describing all possible inventory levels, and (b) sketch the graph of the system.

Short answer

The system of inequalities is: \(A \geq 2B\), \(500A + 700B \leq 30000\), \(A \geq 6\), \(B \geq 3\). The graph is a 2D region bounded by the lines representing these inequalities.

Step-by-step solution

Formulate the equations

Firstly, it's known that model A kayaks need to be twice the number of model B kayaks. This can be written as \(A \geq 2B\). Secondly, due to cost limitations, the total cost should not exceed $30,000, which gives us $500A + $700B \leq 30000. Lastly, they need at least six model A kayaks and three model B kayaks in inventory, giving us two more inequalities: \(A \geq 6\) and \(B \geq 3\). Combining all these we have the system of inequalities: \(A \geq 2B\), \(500A + 700B \leq 30000\), \(A \geq 6\), \(B \geq 3\).

Draw the first inequality

The inequality \(A \geq 2B\) is a straight line passing through origin which divides the plane on one side that contains all the possible solutions. Draw the corresponding area that fulfills this inequality on the graph.

Draw the second inequality

Next, plot \(500A + 700B \leq 30000\) on the same graph. It will be a straight line. Because we're working with \('\leq'\), we want to shade the area under the line, yet still inside the area from the first inequality.

Draw the third and fourth inequality

Finally, plot \(A \geq 6\) and \(B \geq 3\). These will be vertical and horizontal lines respectively, again, make sure to shade the correct area. The area that is shaded on all four inequalities is the solution to the system of inequalities.

Finalize the graph

Now bring together steps 2-4 to complete the graph representing the system of inequalities. Make sure to include labels for all boundaries and shaded areas.

Key concepts

Linear Programming

Linear programming is a mathematical technique used to optimize a certain quantity given several constraints, which are represented by inequalities. In the context of inventory management, linear programming helps to determine the optimal inventory levels so that costs are minimized or profits are maximized.

Consider a store managing its kayak inventory. The goal is to balance between meeting customer demand and keeping inventory costs below a specific limit. Linear programming involves setting up a system of inequalities to model these real-world constraints.

For instance, suppose the store must have at least twice as many model A kayaks as model B kayaks, and the total kayak cost cannot exceed $30,000. These requirements form inequalities that we can solve using linear programming techniques. The solution gives the best mix of kayak models, meeting both the stock and budget constraints without exceeding them.

Graphing Inequalities

Graphing inequalities is an essential skill in visualizing solutions to a system of inequalities. It allows you to see the feasible region where all constraints are satisfied.

To graph inequalities, start by treating each inequality as an equation to find its corresponding line on the graph. Once you have the line, decide which side of the line includes the solutions by selecting a test point. If the test point satisfies the inequality, shade that region of the graph. If not, shade the opposite side.

In the kayak store example, you'll encounter lines corresponding to each inequality like those defining minimum kayak requirements or budgetary constraints. By plotting all these on a graph, the overlapping shaded area represents the set of all possible solutions. Carefully label each boundary and shaded region to clearly delineate the feasible set.

Inventory Management

Inventory management is the strategic control of goods and materials to ensure the right quantity is available without incurring unnecessary costs. For businesses selling products with varying demands and costs, it is vital to balance inventory levels to satisfy both operational and financial goals.

Consider the kayak shop scenario again. The store aims to keep an optimal number of each kayak model to meet demand while sticking to budget constraints. Effective inventory management involves maintaining a stock of at least 6 model A kayaks and 3 model B kayaks, aligned with customer needs and budgetary limits. At the same time, the store must manage its total inventory value, not exceeding $30,000 at any given time.

By applying inventory management techniques, businesses can improve turnover rates, reduce wasted investment in unsold stock, and increase overall efficiency. This, in turn, leads to better customer satisfaction and profitability.