Question

The cost of producing \(x\) units of a product is modeled by \(C=500+300 x-300 \ln x, \quad x \geq 1\) (a) Find the average cost function \(\bar{C}\). (b) Analytically find the minimum average cost. Use a graphing utility to confirm your result.

Short answer

The average cost function is \(\bar{C}(x) = 500/x + 300 - 300\ln{x}/x\) and the minimum average cost is for \(x = exp(1)\) giving the minimum cost of \(\bar{C}(exp(1)) = $500/exp(1) + 300 - 300\)

Step-by-step solution

Compute the Average Cost Function \(\bar{C}(x)\)

The average cost \(\bar{C}(x)\) is the total cost \(C(x)\) divided by the number of units \(x\). Thus, \(\bar{C}(x) = C(x)/x = (500 + 300x - 300\ln{x}) / x = 500/x + 300 - 300\ln{x}/x\).

Compute the derivative of the Average Cost Function

Differentiating the average cost function \(\bar{C}(x)\) with respect to \(x\) gives us \(\bar{C}'(x) = -500/x^2 - 300/(x^2) + 300/x\). This will allow us to find critical points.

Find the Minimum Average Cost

Setting the derivative equal to zero and solving for \(x\) gives us: \(-500/x^2 - 300/(x^2) + 300/x = 0\) which simplifies to \(x = exp(1)\), thus the minimum average cost is \(\bar{C}(exp(1)) = $500/exp(1) + 300 - 300.$

Key concepts

Calculus

Calculus is a branch of mathematics that deals with change and motion. It is essential for analyzing and understanding functions such as the cost functions encountered in economics and optimization problems. A cost function, as seen in the original exercise, is a way to model how the total cost varies with changes in production levels.
To find an average cost function, like the one described in the problem, calculus allows us to differentiate this function, providing insights into how each additional unit produced affects the total cost.
Key concepts of calculus used in cost analysis include:

• **Differentiation** - Finding the derivative of the cost function helps determine the rate of change of cost relative to production, which is crucial when seeking cost efficiency.
• **Critical Points** - Once we differentiate, we can find critical points by setting the derivative to zero, which tells us where costs might be minimized or maximized.

By differentiating the average cost function, \( \bar{C}(x) = \frac{500}{x} + 300 - \frac{300\ln{x}}{x} \,\), we can explore how costs behave as production levels change, leading us to the minimum cost through solving for critical points.

Optimization

Optimization involves finding the most efficient or cost-effective solution while meeting all necessary constraints. In this context, we look at optimizing the average cost of production. The goal is to minimize costs so the business can operate efficiently.
Finding the minimum average cost involves:

• **Deriving the Average Cost Function** - We have already calculated the function, which is key to understanding how costs scale with production.
• **Setting the Derivative to Zero** - This process finds potential minimum or maximum cost points, essential for deciding optimal production levels.

By solving \( \bar{C}'(x) = -\frac{500}{x^2} - \frac{300}{x^2} + \frac{300}{x} = 0 \), we found \( x = e \), where \( e \) is the base of natural logarithms, approximately equal to 2.718.
This value reveals the production level that results in the minimum average cost, highlighting an optimal operation point for the business to consider.
Confirming with a graphing utility further ensures this theoretical value leads to minimal costs.

Cost Analysis

Cost analysis is a critical part of business strategy, allowing firms to understand and manage their expenses. An average cost function is vital in this process as it captures the typical cost of producing one unit over given quantities.
In the given exercise, the total cost function \( C(x) = 500 + 300x - 300\ln{x} \) is divided by \( x \) to yield the average cost function \( \bar{C}(x) \).
From a cost analysis perspective:

• **Evaluating Costs** - Breaking down costs helps businesses understand fixed versus variable expenses, providing insight for pricing strategies.
• **Determining Scale Economies** - The average cost curve can show how costs decrease with increased production, a phenomenon known as economies of scale.
• **Decision Making** - Knowing minimum costs informs decisions about production levels and can guide businesses towards efficient practices.

With the problem showing \( \bar{C}(e) = \frac{500}{e} + 300 - 300 \), cost analysis suggests this is the least amount per unit the firm can hope to achieve, underpinning strategic decisions such as expansion or pricing adjustments.