Problem 83 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The average cost/disc in dollars incurred by Herald Records in pressing \(x\) DVDs is given by the average cost function $$ \bar{C}(x)=2.2+\frac{2500}{x} $$ Evaluate \(\lim _{x \rightarrow \infty} \bar{C}(x)\) and interpret your result.

Step-by-Step Solution

Answer

The limit of the average cost function as the number of DVDs goes to infinity is \(2.2\). As the number of DVDs produced increases, the average cost per disc will approach \(2.20. This means that the company's average cost per disc will settle around \)2.20 as they produce more DVDs.

Step 1: Write down the limit we want to evaluate

We want to find the limit of the average cost function as the number of DVDs goes to infinity: \[\lim_{x \rightarrow \infty} \bar{C}(x).\]

Step 2: Separate the function into a constant term and a rational term

The function \(\bar{C}(x)\) has two terms, a constant term, and a rational term. We can rewrite the function as: \[\bar{C}(x)= 2.2 + \frac{2500}{x}.\]

Step 3: Evaluate the limit of each term separately

Using the properties of limits, we can evaluate the limit of the sum of the two terms as the sum of the individual limits: \[\lim_{x \rightarrow \infty} \bar{C}(x) = \lim_{x \rightarrow \infty} ( 2.2) + \lim_{x \rightarrow \infty} (\frac{2500}{x}).\] The first limit is a constant, so its limit will be the constant itself: \[\lim_{x \rightarrow \infty} 2.2 = 2.2.\] For the second limit, as \(x\) goes to infinity, the fraction \(\frac{2500}{x}\) goes to \(0\). Thus: \[\lim_{x \rightarrow \infty} (\frac{2500}{x})=0.\]

Step 4: Combine the limits

Now combine the limits we found in Step 3 to get the overall limit: \[\lim_{x \rightarrow \infty} \bar{C}(x) = 2.2 + 0 = 2.2.\] So, the limit of the average cost function as the number of DVDs goes to infinity is \(2.2\). Interpretation: As the number of DVDs produced approaches infinity, the average cost per disc will approach \(2.20. This means that, as the company produces more and more DVDs, the average cost per disc will settle around \)2.20.

Key Concepts

Average Cost Function

The average cost function is a mathematical way to determine the cost per item when producing multiple units of the same product. In our exercise, this function tells us the cost per DVD in pressing DVDs for Herald Records. Specifically, it is given by the equation:\[ \bar{C}(x) = 2.2 + \frac{2500}{x} \]Here, \(x\) represents the number of DVDs produced, while \(\bar{C}(x)\) gives the average cost per DVD. The first term, 2.2, is a fixed cost component, meaning no matter how many DVDs are made, this cost remains constant. The second term, \(\frac{2500}{x}\), varies based on the quantity produced. As production increases, this second part of the function decreases, reflecting economies of scale.

Rational Function

A rational function is an expression

• consisting of a ratio
• of two polynomials.

In our case, \(\frac{2500}{x}\) is the rational term in the average cost function. This term is particularly interesting: when we look at its behavior as \(x\) becomes very large (approaching infinity), it tends toward zero. That’s because dividing a constant by a larger and larger number results in a smaller outcome. The rational function here highlights the decreasing impact of fixed overhead costs when production scales up. This concept is pivotal in understanding how large-scale production can reduce per-unit costs.

Cost Analysis

Cost analysis involves breaking down costs to understand financial growth, profit margins, and organizational efficiency. Through the average cost function, we analyze how costs behave with changes in production levels. The analysis reveals two primary insights:- **Fixed Costs**: Fixed costs, such as 2.2 in our equation, remain unchanged regardless of production quantity.- **Variable Costs**: The variable component \(\frac{2500}{x}\) reflects costs that vary with production. This part decreases as \(x\) increases, reducing the overall cost per DVD.By evaluating the limit, we show that at very high production levels, the average cost stabilizes, providing valuable information for strategic planning and pricing strategies.

Applied Mathematics

Applied mathematics uses mathematical methods and models to solve real-world problems. In the context of cost analysis, it provides a framework to analyze business operations efficiency. By evaluating \[ \lim _{x \rightarrow \infty} \bar{C}(x) \]the function

• demonstrates the concept of limits
• as a tool to predict long-term trends.

This helps businesses anticipate how costs behave under different production scenarios. As production scales, understanding limits allows companies to maximize efficiency and minimize costs, thereby optimizing profit margins. Such analyses are crucial for competitive industries where keeping costs low is a significant advantage.