Problem 66 - Calculus an Applied Approach

Question

Inventory Replenishment The ordering and transportation cost \(C\) per unit for the components used in manufacturing a product is \(C=\left(375,000+6 x^{2}\right) / x, \quad x \geq 1\) where \(C\) is measured in dollars and \(x\) is the order size. Find the rate of change of \(C\) with respect to \(x\) when \((a) x=200,\) (b) \(x=250,\) and (c) \(x=300 .\) Interpret the meaning of these values.

Step-by-Step Solution

Answer

The rates of change at \(x=200\), \(x=250\), and \(x=300\) can be calculated using the derivative of the cost function. The value of the derivative at these points reveals whether the cost is increasing or decreasing at those order sizes.

Step 1: Differentiate the Cost Function

Differentiate the function \(C=\left(375,000+6 x^{2}\right) / x\). Use power rule and quotient rule steps as applicable to get the derivative \(\frac{dC}{dx}\).

Step 2: Find Rate of Change at x=200

Substitute \(x=200\) in the derivative \(\frac{dC}{dx}\) obtained from step 1. This will give the rate of change of cost at this order size.

Step 3: Find Rate of Change at x=250

Substitute \(x=250\) in the derivative \(\frac{dC}{dx}\). This will give the rate of change of cost at this order size.

Step 4: Find Rate of Change at x=300

Substitute \(x=300\) in the derivative \(\frac{dC}{dx}\). This will give the rate of change of cost at this order size.

Step 5: Interpret the results

Negative rate of change means the cost is decreasing as the order size increases, while a positive rate signifies an increasing cost.

Key Concepts

Inventory Replenishment

Inventory replenishment is an essential part of supply chain management. It involves restocking the inventory to ensure that there is a sufficient supply of products or components available to meet customer demand or production needs. In mathematical terms, we analyze how changes in order size can impact costs. In this context, the cost per unit is given by the formula \(C=\frac{375,000+6 x^{2}}{x}\), where \(C\) is measured in dollars and \(x\) is the order size.

• Understanding this cost function helps businesses plan their inventory purchases efficiently.
• The primary goal is to minimize costs while maintaining adequate inventory levels.

Effectively managing inventory replenishment requires analyzing the cost function, which leads us to concepts like differentiation to find out how cost changes as order size varies.

Differentiation

Differentiation in calculus is the process of finding the rate at which a function changes at any given point. For our inventory cost function \(C\), we want to determine \(\frac{dC}{dx}\), the derivative, to understand how the cost per unit changes with varying order sizes.

• This derivative tells us the sensitivity of the cost to changes in order size.
• It answers how quickly and in what direction (increasing or decreasing) the cost changes.

In practical terms, differentiation can help a business decide if increasing order size will save money or if smaller orders might be more cost-effective.

Rate of Change

The rate of change is a concept that describes how one quantity changes in relation to another. For our exercise, the rate of change refers to \(\frac{dC}{dx}\), which quantifies how the cost per unit \(C\) changes as the order size \(x\) changes.

• A positive rate of change means that as the order size increases, the cost also increases.
• A negative rate of change indicates that an increase in order size leads to a decrease in cost.

Understanding the rate of change allows businesses to make informed decisions about order sizes, helping them find a balance between cost efficiency and inventory need.

Quotient Rule

The quotient rule is a technique used in differentiation when dealing with functions that are divided by each other. In our exercise, the cost function \(C\) is represented as a quotient \(\frac{375,000+6 x^{2}}{x}\). To differentiate it, we apply the quotient rule which states:
\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\]where \(u = 375,000+6x^2\) and \(v = x\).

• This method enables us to handle more complex differentiation involving ratios effectively.
• Using the quotient rule helps in accurately determining how each component of the function impacts the overall rate of change.

By utilizing the quotient rule, businesses can further refine their analysis of costs related to inventory replenishment.

Power Rule

The power rule is a fundamental tool in calculus for differentiating functions of the form \(x^n\). It simplifies the differentiation process by reducing it to a straightforward procedure: if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). In the context of our problem, the term \(6x^2\) within the cost function \(C\) can be differentiated using the power rule.

• By applying the power rule, differentiating terms like \(6x^2\) resulting in \(12x\).
• The power rule accelerates the differentiation process for polynomial terms, making complex functions more manageable.

When analyzing inventory costs, the power rule allows for swiftly identifying changes in specific components of the cost function, leading to better decision-making regarding inventory orders.