Question

The cost and revenue functions for a product are \(C=34.5 x+15,000\) and \(R=69.9 x\) (a) Find the average profit function \(\bar{P}=(R-C) / x\). (b) Find the average profits when \(x\) is \(1000,10,000\), and 100,000 (c) What is the limit of the average profit function as \(x\) approaches infinity? Explain your reasoning.

Short answer

The average profit function is \( \bar{P} = 35.4 - 15000/x \). The average profits for x=1000, 10000, and 100000 are respectively 20.4, 33.9, and 35.25. As x approaches infinity, the average profit per unit will stabilize at $35.4.

Step-by-step solution

Derive the average profit function

The profit function is simply the revenue function minus the cost function. Dividing by x gives the average profit per unit. Thus the average profit function \(\bar{P}\) is: \(\bar{P}=(R-C) / x = \((69.9x-(34.5 x+15,000)) / x\)

Simplifying the average profit function

Algebraically simplify the average profit function: \(\bar{P}=(69.9x-34.5x-15000) / x = (35.4x-15000)/x = 35.4 - 15000/x\)

Calculate the average profits for x = 1000, 10000, and 100000

For \(x=1000\), substitute into the average profit function: \( \bar{P} = 35.4 - 15000/1000 = 35.4 - 15 = 20.4 \). For \(x=10000\), substitute into the average profit function: \( \bar{P} = 35.4 - 15000/10000 = 35.4 - 1.5 = 33.9 \). For \(x=100000\), substitute into the average profit function: \( \bar{P} = 35.4 - 15000/100000 = 35.4 - 0.15 = 35.25 \).

Find the limit of the average profit as x approaches infinity.

The limit of the average profit function as \(x\) approaches infinity is found by using the limit rules. As \(x\) becomes infinitely large, the fraction \(15000/x\) approaches 0. Therefore, the limit of \(\bar{P}\) as \(x\) approaches infinity is \(35.4 - 0 = 35.4\). This indicates that as the quantity of units produced gets infinitely large, the average profit per unit tends toward $35.4. This means that the profit per unit will stabilize at $35.4 when production increases indefinitely.

Key concepts

Limit of a Function

Understanding the limit of a function is a fundamental concept in calculus, and it can be particularly insightful in economics. When we talk about limits in mathematics, we're referring to the behavior of a function as the input approaches a certain value. In the context of the average profit function, the exercise asks about the limit as the number of goods produced, represented by x, approaches infinity.

Why is this important for students of economics? Because it tells us what happens to average profit when production scales up to very large quantities. This can inform business strategies for long-term growth or understanding market saturation points. For our average profit function \( \bar{P} = 35.4 - \frac{15000}{x} \), as x becomes very large, the term \( \frac{15000}{x} \) becomes negligible, effectively approaching zero. Therefore, the limit of \( \bar{P} \) as x approaches infinity is the constant term, 35.4, suggesting that average profit stabilizes at this value regardless of further increases in production volume.

Profit Maximization

The goal of profit maximization is foundational in business and economics. It's the process of finding the most efficient operating level where a company can achieve the highest profit. In theoretical models, profit maximization can be understood through the analysis of cost and revenue functions. The profit function \( P \) equals total revenue \( R \) minus total cost \( C \). By finding the average profit function and analyzing its behavior especially by looking at its limits, as previously discussed, a company can estimate the most profitable scale of operations.

The average profit \( \bar{P} \) is particularly useful because it allows businesses to evaluate profitability on a per-unit basis, which can be more insightful than looking at total profit alone. High average profits at certain quantities might suggest an optimal range for production and sales, guiding decisions such as resource allocation, pricing strategy, and market expansion.

Algebraic Simplification

When dealing with complex mathematical expressions, algebraic simplification is a useful skill to make these expressions more manageable and easier to understand. Simplification can involve factoring, expanding expressions, reducing fractions, or canceling out terms. It's like cleaning up a mess: once you clear the clutter, it's easier to see what you have to work with. For example, take the average profit function from the exercise: \( \bar{P} = (69.9x - (34.5x + 15000)) / x \). By simplifying this function to \( \bar{P} = 35.4 - \frac{15000}{x} \), we arrived at a much more straightforward form. That simplified form allows us to more directly see the relationship between production volume and average profit, as well as to easily compute the limit as production volume increases.

Remember, simplification doesn't change the value of the expression; it only makes it easier to work with, whether you're solving equations, derivative taking, integrating, or just trying to grasp what the numbers are telling you.