Question

The number of units in inventory in a small company is \(N=25\left(2 \| \frac{t+2}{2} \rrbracket-t\right), \quad 0 \leq t \leq 12\) where the real number \(t\) is the time in months. (a) Use the greatest integer function of a graphing utility to graph this function, and discuss its continuity. (b) How often must the company replenish its inventory?

Short answer

The inventory needs to be replenished whenever the function N(t) hits zero. After graphing and analyzing the function, you can count these points to find the number of replenishments needed in 12 months.

Step-by-step solution

Understanding the Floor Function

The floor function, denoted as \(\lfloor x \rfloor\), gives the greatest integer less than or equal to \(x\). For instance, \(\lfloor 2.3 \rfloor=2\) and \(\lfloor -2.3 \rfloor=-3\). In the given equation, the floor function is applied to the quantity \((t+2)/2\).

Plot the Function

Plot the equation \(N=25\left(2 \lfloor \frac{t+2}{2} \rfloor -t\right)\) for \(0 \leq t \leq 12\). Due to the floor function, the graph will look continuous but have 'jumps' at certain points. At these points, the function is not technically continuous.

Analyze the Graph

Look for the points where the graph decreases rapidly to 0. These are the points where the inventory drops to zero and needs to be replenished.

Count Replenishments

Count the number of times the graph hits zero (inventory depletion points). This gives the number of times the company needs to replenish its inventory within the year.

Key concepts

Inventory Management

Understanding inventory management is key to ensuring smooth operations of a company. In this exercise, we explore an inventory model through a mathematical expression. It helps predict when inventory runs out and needs replenishment. This is crucial for maintaining product flow without overstocking or shortages.

In our equation, the variable \(N\) represents the units of inventory left in months. This modeling can help businesses plan inventory restocking schedules. Proper management ensures businesses meet demand and reduces holding costs.

Maintaining optimal inventory levels prevents lost sales or excess storage costs. This exercise demonstrates the importance of integrating mathematical modeling in inventory strategies.

Continuity of Functions

Continuity of functions is a fundamental concept in mathematics that helps us understand how smooth a function is across its domain. Some functions can be broken or 'jump' at certain points, making them discontinuous.

In this exercise, the function involves the greatest integer function, which results in these types of jumps. Though it appears continuous, it's not technically so at points where the inventory is replenished.

This happens because the greatest integer function introduces segment changes at various points. This loss of continuity is crucial for identifying moments when the inventory reaches zero.

• The greatest integer function causes the jumps
• Discontinuity moments indicate a need for inventory replenishment

Graphical Analysis

Graphical analysis is a powerful tool that allows us to visualize and better understand the behavior of a function. In this scenario, graphical methods help identify trends and disruptions in inventory levels over time.

By plotting the given equation \(N=25(2\lfloor \frac{t+2}{2} \rfloor -t)\) for \(0 \leq t \leq 12\), you can easily notice sharp drops in inventory at specific times. These are moments when the greatest integer function causes jumps, signaling a reset in inventory.

Through graphical analysis, businesses can anticipate demand and plan replenishment schedules. Visualizations make complex data more accessible and actionable. It also helps in identifying the essential points where stock levels need boosting to prevent zero stockouts.