In the world of manufacturing, understanding cost functions is crucial for making informed decisions about production levels. The function c(x) = 0.7x² - 238x + 30,789 is a quadratic equation that helps determine the cost associated with producing x number of machines. This article will explore this function in depth, providing insights into its components, how to solve for x, and the implications for production strategies.
The primary goal of this article is to analyze the function and find out how many machines must be produced to meet specific cost criteria. By breaking down the components of the function and applying mathematical techniques, we can provide a comprehensive understanding of the cost implications of manufacturing decisions.
As we dive into the details of this cost function, readers will find practical applications of mathematical concepts in real-world scenarios. We will also discuss the importance of understanding such functions for effective business planning and resource allocation.
Table of Contents
1. Introduction to the Cost Function
The cost function c(x) = 0.7x² - 238x + 30,789 represents the total cost incurred in producing x machines. This function is essential for manufacturers as it helps them determine the financial implications of scaling production. Understanding this function is vital for making informed business decisions.
2. Function Analysis: Breaking Down c(x)
2.1 Components of the Function
This quadratic function consists of three main components:
- 0.7x²: This term represents the variable costs that increase with the square of the number of machines produced. It indicates that as production increases, the cost increases at an accelerating rate.
- -238x: This linear term accounts for the fixed costs that decrease as more machines are produced. It represents economies of scale.
- 30,789: This constant term represents the initial fixed costs incurred regardless of the production level.
2.2 Graphing the Function
Visualizing the cost function is crucial for understanding its behavior. The graph of the function c(x) = 0.7x² - 238x + 30,789 is a parabola that opens upwards.
In the graph, the x-axis represents the number of machines produced, while the y-axis represents the total cost. The vertex of the parabola indicates the minimum cost point, which is essential for manufacturers to identify.
3. Solving the Equation for x
3.1 Using the Quadratic Formula
To find the number of machines x that should be produced to achieve a specific cost, we can rearrange the equation and apply the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
For our function, a = 0.7, b = -238, and c = 30,789.
3.2 Calculating the Number of Machines
Let’s say we want to determine how many machines must be produced to reach a cost of, for example, $50,000.
We set c(x) = 50,000:
0.7x² - 238x + 30,789 = 50,000
Rearranging gives us:
0.7x² - 238x - 19,211 = 0
Now we can apply the quadratic formula:
x = (238 ± √((-238)² - 4 * 0.7 * (-19,211))) / (2 * 0.7)
Calculating the discriminant and solving for x will give us the number of machines required.
4. Real-World Implications of the Function
Understanding the implications of the cost function is vital for effective business operations. By analyzing c(x), manufacturers can make decisions about scaling production, pricing strategies, and resource allocation. Here are some key points:
- Identifying the optimal production level to minimize costs.
- Understanding economies of scale and how they affect profitability.
- Making informed pricing decisions based on production costs.
5. Conclusion
In conclusion, the function c(x) = 0.7x² - 238x + 30,789 is a powerful tool for manufacturers to understand their production costs. By solving for x, businesses can determine the number of machines to produce to achieve specific cost objectives. Understanding such functions is essential for making informed decisions and optimizing operations.
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