Calculating Profit Mike Makes From 4 Custom Tables F(x) And G(x)
Introduction
In this article, we will delve into the mathematical functions that represent Mike's earnings and costs associated with making custom tables. The function f(x) = 60x + 4 models Mike's earnings, where 'x' represents the number of tables he makes. This linear function indicates that Mike earns a base amount of $4, plus an additional $60 for each table he completes. On the other hand, the function g(x) = 2x - 4 represents Mike's upfront costs, with 'x' again denoting the number of tables. This function suggests that Mike incurs a cost of $2 per table, but also benefits from a cost reduction of $4, possibly due to initial setup or material discounts. Our primary objective is to determine Mike's profit from making four tables, which can be achieved by calculating (f - g)(4). This involves subtracting the total cost function g(x) from the total earnings function f(x) and then evaluating the result for x = 4. This calculation will provide us with the net profit Mike makes after accounting for his costs. Understanding these functions and their application is crucial for Mike to effectively manage his business and make informed decisions about pricing and production. We will break down each step of the calculation, ensuring clarity and a comprehensive understanding of the process. This analysis will not only provide a numerical answer but also a broader insight into how mathematical functions can be used to model and analyze real-world business scenarios.
Understanding the Functions
Before we dive into the calculation, let's thoroughly understand the functions at play. The function f(x) = 60x + 4 is a linear function that represents Mike's earnings from making custom tables. The coefficient 60 signifies that Mike earns $60 for each table he makes. This is the variable cost component of his earnings, directly proportional to the number of tables. The constant term +4 represents a fixed earning, irrespective of the number of tables made. This could be a base fee, a setup charge, or any other fixed income Mike receives. Understanding this breakdown allows us to appreciate how each table contributes to Mike's earnings and the baseline income he can expect. Conversely, the function g(x) = 2x - 4 represents Mike's upfront costs. The coefficient 2 indicates that Mike incurs a cost of $2 for each table he makes. This could be the cost of materials, labor, or other variable expenses that increase with the number of tables. The constant term -4 represents a cost reduction of $4. This could be due to discounts on bulk purchases, initial setup subsidies, or any other factor that reduces his costs. It's important to note that this cost reduction is a one-time benefit and doesn't scale with the number of tables. Analyzing these two functions, we can see how Mike's earnings and costs are structured. The difference between these functions will ultimately determine his profit, which is the core of our analysis. By understanding the individual components of these functions, we can gain a deeper understanding of Mike's business model and the factors influencing his profitability. This foundational knowledge is essential for accurately calculating and interpreting Mike's profit from making custom tables.
Calculating (f - g)(x)
To determine Mike's profit, we first need to find the profit function, which is represented by (f - g)(x). This involves subtracting the cost function g(x) from the earnings function f(x). Mathematically, this is expressed as (f - g)(x) = f(x) - g(x). We have f(x) = 60x + 4 and g(x) = 2x - 4. Substituting these into the equation, we get: (f - g)(x) = (60x + 4) - (2x - 4). Now, we need to simplify this expression. We start by distributing the negative sign in front of the parentheses: (f - g)(x) = 60x + 4 - 2x + 4. Next, we combine like terms. We have two terms with 'x': 60x and -2x, which combine to 58x. We also have two constant terms: 4 and 4, which combine to 8. Therefore, the simplified profit function is: (f - g)(x) = 58x + 8. This function represents Mike's profit for making 'x' tables. The coefficient 58 indicates that Mike makes a profit of $58 for each table after accounting for the costs. The constant term +8 represents a fixed profit, irrespective of the number of tables made. This could be due to the initial cost reduction or any other fixed income. Understanding the profit function is crucial for analyzing Mike's business. It allows him to quickly calculate his profit for any number of tables and to understand the factors influencing his profitability. The positive coefficient of 'x' indicates that his profit increases with the number of tables made, while the constant term provides a baseline profit even if no tables are made. This profit function is a powerful tool for Mike to manage his business and make informed decisions.
Evaluating (f - g)(4)
Now that we have the profit function (f - g)(x) = 58x + 8, we can determine Mike's profit from making four tables by evaluating (f - g)(4). This means substituting x = 4 into the profit function. So, we have: (f - g)(4) = 58(4) + 8. First, we perform the multiplication: 58 * 4 = 232. Then, we add the constant term: 232 + 8 = 240. Therefore, (f - g)(4) = 240. This result tells us that Mike will make a profit of $240 from making four custom tables. This profit accounts for both his earnings and his upfront costs. The $240 profit is a significant amount, indicating that Mike's business is potentially profitable. The majority of this profit comes from the variable profit of $58 per table, while a smaller portion comes from the fixed profit of $8. This breakdown helps us understand the factors contributing to Mike's profit. The profit of $240 can be used by Mike for various purposes, such as reinvesting in his business, paying himself a salary, or saving for future expenses. It's important to note that this profit is calculated based on the given functions, which may not fully represent all aspects of Mike's business. There may be other costs or earnings that are not included in these functions. However, this calculation provides a valuable estimate of Mike's profit and can be used as a basis for further analysis. Understanding the profit from making four tables is a crucial step in evaluating the financial viability of Mike's business. It provides a concrete number that can be used to assess the success of his business and to make informed decisions about future operations.
Conclusion
In conclusion, by analyzing the functions representing Mike's earnings and costs, we have successfully determined his profit from making four custom tables. The function f(x) = 60x + 4 models Mike's earnings, while the function g(x) = 2x - 4 represents his upfront costs. By calculating (f - g)(4), we found that Mike will make a profit of $240 from making four tables. This calculation involved several steps: first, we understood the individual functions and their components; second, we calculated the profit function (f - g)(x) by subtracting g(x) from f(x); and third, we evaluated (f - g)(4) by substituting x = 4 into the profit function. The result of $240 represents the net profit Mike makes after accounting for his costs. This profit is a crucial indicator of the financial health of Mike's business. It can be used to assess the viability of his business model, to make informed decisions about pricing and production, and to plan for future growth. The analysis of these functions provides a valuable framework for understanding Mike's business and for making sound financial decisions. The profit of $240 can be used for various purposes, such as reinvesting in the business, paying salaries, or saving for future expenses. This analysis also highlights the importance of mathematical functions in modeling and analyzing real-world business scenarios. By using functions to represent earnings and costs, we can gain a deeper understanding of the factors influencing profitability and make more informed decisions. This approach can be applied to various business situations and is a valuable tool for entrepreneurs and business managers.
Find (f-g)(4) given f(x) = 60x + 4 represents earnings and g(x) = 2x - 4 represents costs, to determine the profit from 4 tables.
Calculating Profit Mike Makes from 4 Custom Tables f(x) and g(x)
