Question

A business has a cost (in dollars) of \(C=0.5 x+500\) for producing \(x\) units. (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=250\) and when \(x=1250\). (c) What is the limit of \(\bar{C}\) as \(x\) approaches infinity?

Short answer

The average cost function is \(\bar{C}(x) = 0.5 + 500/x\). When \(x=250\), \(\bar{C} = 2\) and when \(x=1250\), \(\bar{C} = 0.9\). The limit of the average cost as \(x\) approaches infinity is \(0.5\).

Step-by-step solution

Average Cost Function

To find the average cost function \(\bar{C}\), divide the cost function \(C(x)\) by \(x\). So, \(\bar{C}(x) = C(x)/x = (0.5 x + 500)/x = 0.5 + 500/x\).

Find \(\bar{C}\) at Specific Values

Plug the specific values of \(x=250\) and \(x=1250\) into the average cost function, giving \(\bar{C}(250) = 0.5 + 500/250 = 2\) and \(\bar{C}(1250) = 0.5 + 500/1250 = 0.9\).

Limit of \(\bar{C}\) as \(x\) Approaches Infinity

The limit of a function as \(x\) approaches infinity describes the behavior of the function when \(x\) becomes very large. As \(x\) goes to infinity, \(500/x\) goes to \(0\). So, \(\lim_{x\to\infty} \bar{C}(x) = \lim_{x\to\infty}(0.5 + 500/x) = 0.5 + 0 = 0.5\).

Key concepts

Cost Function

A cost function is used to describe how much it costs to produce a given number of goods or services. It expresses the total cost, which typically includes fixed and variable costs for producing a product. Fixed costs are expenses that do not change with the level of output, such as rent or salaries. Variable costs, however, do change according to the number of items produced, like raw materials.
Example of a Simple Cost Function:

• The cost function provided in the exercise is given as \(C = 0.5x + 500\).
• Here, \(0.5x\) represents the variable cost, dependent on each additional unit produced.
• The constant 500 is a fixed cost; it remains the same regardless of how many units are produced.

This understanding of cost functions is crucial for businesses to determine pricing, budgeting, and profitability strategies. By analyzing cost functions, a company can decide the optimal number of goods to produce to achieve the lowest cost per unit and maximize profit.

Limit at Infinity

When we talk about the limit of a function as \(x\) approaches infinity, we're interested in what happens to the function's value when \(x\) becomes very large. This concept helps us understand the long-term behavior of mathematical models or phenomena.
The Limit for the Average Cost Function:

• For the average cost function in the exercise, \(\bar{C} = 0.5 + \frac{500}{x}\), we analyze \(\lim_{x\to\infty}\bar{C}(x)\).
• As \(x\) increases indefinitely, \(\frac{500}{x}\) approaches zero because 500 divided by a very large number is almost nothing.
• Thus, the limit \[\lim_{x\to\infty} (0.5 + \frac{500}{x}) = 0.5.\]

This result indicates that as more and more units are produced, the average cost per unit approaches 0.5 units of currency, demonstrating economies of scale in production.

Calculus Application

Calculus is a branch of mathematics that helps us understand change. It is incredibly useful for solving problems involving rates of change and quantities under movement or progression. In business and economics, calculus is often used to optimize production, minimize costs, or maximize profit.
Average Cost Functions and Calculus:

• Calculus allows us to derive an average cost function from a total cost function. This process involves mathematical operations like differentiation and integration.
• The average cost function \(\bar{C}(x) = \frac{C(x)}{x}\) is derived by dividing the total cost by the number of units \(x\).
• Calculus also allows us to examine limits, which helps determine long-term behaviors and optimal production quantities.

By applying these methods, businesses can simplify complex cost structures, making them easier to analyze and understand. This empowers decision makers to better control expenses and enhance financial performance.

Business Mathematics

Business mathematics is an essential field at the crossroad of mathematics and business management. It provides tools and techniques to solve business-related problems through mathematical principles.
Applications in Business and Economics:

• Cost functions, like the one given, help businesses understand their financial commitments relative to production levels.
• Average costs derived from these functions assist in pricing strategies, allowing companies to determine selling prices that cover costs and deliver profitability.
• Mathematics in business also encompasses revenue analysis, profit maximization, and risk assessment.

Understanding these mathematical aspects allows business leaders to make informed decisions. They can evaluate profitability, predict financial performance, and strategically plan production operations. Thus, business mathematics functions as the backbone of rational and effective financial management.