Question

The cost function for a company to recycle \(x\) tons of material is given by \(C=1.25 x+10,500\), where \(C\) is measured in dollars. (a) Find the average cost function \(\bar{C}\). (b) Find the average costs of recycling 100 tons of material and 1000 tons of material. (c) Determine the limit of the average cost function as \(x\) approaches infinity. Interpret the limit in the context of the problem.

Short answer

First, we find the average cost function by dividing the given cost function by \(x\), getting \(\bar{C} = \frac{1.25x + 10500}{x}\). Then the average costs for recycling 100 tons and 1000 tons can be found by substituting \(x\) with 100 and 1000 respectively in \(\bar{C}\). Lastly, the limit of the average cost function as \(x\) approaches infinity can be found by calculating \(\lim_{x \to \infty} \bar{C}\), which gives the response of the average cost function to very large quantities of material.

Step-by-step solution

Identifying The Average Cost Function

The average cost function \(\bar{C}\) is obtained by dividing the total cost \(C\) by the quantity \(x\). So the average cost function can be expressed as \(\bar{C} = \frac{C}{x}\). Here the cost function \(C\) is given by \(1.25x + 10500\).

Calculation of Average Costs For Specific Material Tons

According to the formula of average cost function we have calculated previously, we can calculate the average costs for 100 tons and 1000 tons by replacing \(x\) with the required quantity. So for \(x=100\) tons, we will calculate it as \(\bar{C}_{100} = \frac{1.25*100 + 10500}{100}\) and for \(x=1000\) tons, it will be calculated as \(\bar{C}_{1000} = \frac{1.25*1000 + 10500}{1000}\).

Interpretation of Limit of Average Cost Function as \(x\) Approaches Infinity

The limit of any function as \(x\) approaches infinity gives the behavior of the function at very large values of \(x\). We can calculate this as \(\lim_{x \to \infty} \bar{C} = \lim_{x \to \infty} \frac{1.25x + 10500}{x}\). The result will furnish the response of the average cost function to recycling very large quantities of material.

Key concepts

Cost Function

A cost function in mathematics, particularly in calculus, is a model that relates the cost incurred by a business to produce a certain quantity of goods to the quantity produced. In the context of a company recycling material, the cost function would encompass all the variables that contribute to the total cost of recycling, which includes fixed costs, like machinery and rent, and variable costs, like labor and materials.

For instance, the cost function provided by the textbook, C=1.25x+10,500, implies that for every ton of material recycled, there is an additional cost of \(1.25, and there is also a fixed cost of \)10,500 associated with recycling, regardless of the quantity. Understanding the components of this function is crucial for businesses as it helps them strategize on pricing, budgeting, and even scaling operations.

Limit of a Function

The concept of a limit is a foundational element in calculus, used to describe the behavior of a function as the input approaches a certain point or tends toward infinity. It finds the value that a function's output approaches, as the input value approaches some value. In the given exercise, the limit of the average cost function as x approaches infinity is of interest.

Calculating this limit helps us understand what happens to the average cost per unit when production levels become very large. It answers questions like 'Does the average cost keep increasing, decrease, or level off as we recycle more and more tons of material?'. This is particularly relevant in economics to assess economies of scale and to make long-term cost predictions.

Calculus in Economics

Calculus, the mathematical study of continuous change, is a powerful tool in economics. It helps economists and businesses model and analyze dynamic conditions. Through the use of derivatives and integrals, they can predict maximum profit or minimum cost, the rate of change of economic quantities, and optimal resource allocation.

For example, in the application to the recycling company's average cost function, calculus allows the determination of how average costs change with quantity. By finding the derivative of the given cost function, we can assess the marginal cost, or the cost of producing one more unit of a good. This is crucial for making effective production and pricing decisions in the company's operations.