Question

The cost and revenue functions for a product are \(C=25.5 x+1000\) and \(R=75.5 x\) (a) Find the average profit function \(\bar{P}=\frac{R-C}{x}\). (b) Find the average profits when \(x\) is 100,500 , and 1000 . (c) What is the limit of the average profit function as \(x\) approaches infinity? Explain your reasoning.

Short answer

The average profit function is \( \bar{P} = 50 - \frac{1000}{x} \). The average profits for 100, 500, and 1000 units are 40, 48, and 49 respectively. The limit of the average profit function as \(x\) approaches infinity is 50.

Step-by-step solution

Formulation of Average Profit Function

The average profit function can be derived as follows: \( \bar{P} = \frac{R-C}{x} \). This is the given formula to find the average profit, which is subtraction of the cost function from revenue function all divided by x. By plugging the given functions into the equation, we get: \( \bar{P} = \frac{75.5x- (25.5x+1000)}{x} = \frac{50x-1000}{x} \). This simplifies to \( \bar{P} = 50 - \frac{1000}{x} \).

Calculation of Average Profits

We can find the average profits for the given quantities by substituting the values into the average profit function. For x=100, \( \bar{P}_{100} = 50 - \frac{1000}{100} = 40 \); For x=500, \( \bar{P}_{500} = 50 - \frac{1000}{500} = 48 \); and For x=1000, \( \bar{P}_{1000} = 50 - \frac{1000}{1000} = 49 \).

Limit of Average Profit Function

The limit as x approaches infinity is found by evaluating the function when x is significantly large. As x approaches infinity, the term \( \frac{1000}{x} \) approaches zero. Thus, the function simplifies to \( \bar{P} = 50 - 0 \). So, the limit as x approaches infinity is 50.

Key concepts

Cost Function

Understanding the cost function is vital in business calculus, as it describes how the total cost to produce a good changes with the level of production. It can be written as a formula, such as in the example exercise where the cost function is given by C = 25.5x + 1000. Here, 25.5x represents the variable costs that increase with the number of units produced, x, while 1000 represents the fixed costs that do not change with the level of production.
When calculating profits or other economic metrics, understanding how to interpret this cost function allows businesses to forecast expenses and set production levels appropriately. It's the starting point for determining the profitability of selling goods at different scales.

Revenue Function

Complementing the cost function, the revenue function shows the total income received from selling a certain amount of goods or services. In the textbook exercise, the revenue function is defined as R = 75.5x. This linear function indicates that revenue increases directly with the quantity x being sold, at a constant rate of 75.5 per unit.
A fundamental concept in business calculus, the revenue function helps in understanding the relationship between pricing, sales volume, and total revenue. By comparing the revenue function with the cost function, businesses can analyze their operations and make informed decisions to maximize profits.

Limit of a Function

The concept of the limit of a function is crucial in calculus and particularly in business applications. A limit describes the behavior of a function as the input (in this case, the number of goods produced and sold, x) approaches a certain value. For example, the exercise asks for the limit of the average profit function as x approaches infinity.
In business calculus, limits can provide insights into long-run behavior of economic functions. By evaluating the limit of the average profit function, we can deduce the expected stable state of profits when production is expanded indefinitely. This is illustrated in the exercise where the average profit tends toward 50 as the quantity produced grows larger, a concept we'll examine further in the context of average profit function.

Business Calculus

Business calculus is a branch of mathematics applying calculus principles to the problems of business and economics. Utilizing functions such as cost, revenue, and profit functions, business calculus helps in making strategic decisions. For instance, determining optimal production levels or understanding trends in cost and revenue requires calculus techniques like differentiation and interpreting limits.
In the given exercise, business calculus is used to derive the average profit function and analyze its behavior as production increases. This analysis ensures that business strategies are data-driven and are grounded in the mathematical realities of the company's operational metrics.