When analyzing the function c(x) = 0.6x² - 168x + 30,389, it’s crucial to understand its significance in determining the minimum unit cost associated with a production process. This quadratic function represents the cost of producing a certain number of units (x) and can provide insights into how costs change as production levels vary. In this article, we will explore how to find the minimum unit cost by analyzing the function step by step.
First, we will delve into the nature of quadratic functions and their graphical representations. Quadratic functions are parabolic in shape, which means they can either open upwards or downwards. The specific function we are discussing opens upwards because the coefficient of x² is positive (0.6). This characteristic indicates that the function has a minimum point, which is essential for calculating the minimum unit cost.
Next, we will also take a look at the key components of this function, including the coefficients and the constant term, and how they influence the overall behavior of the cost function. By the end of this article, you will have a comprehensive understanding of how to derive the minimum unit cost from the given function.
Table of Contents
Understanding Quadratic Functions
Quadratic functions are polynomial functions of degree two, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. In our function c(x), we have:
- a = 0.6
- b = -168
- c = 30,389
The shape of a quadratic function is a parabola. Since our function opens upwards (as a > 0), it has a minimum point known as the vertex. This vertex will provide the minimum unit cost, which is essential for businesses in controlling production costs.
Graphical Representation of the Function
To visualize the function c(x), we can plot it on a graph. The x-axis will represent the number of units produced (x), and the y-axis will represent the cost (c). The curve will show how the cost varies with the number of units produced.
As the graph indicates, there is a point where the cost reaches its minimum value. This point can be located using the vertex formula:
Vertex x-coordinate = -b / (2a)
Finding the Vertex of the Function
To find the vertex of the function c(x) = 0.6x² - 168x + 30,389, we use the vertex formula mentioned above:
Vertex x-coordinate = -(-168) / (2 * 0.6) = 168 / 1.2 = 140
Now that we have the x-coordinate of the vertex, we can substitute this value back into the function to find the corresponding y-coordinate (minimum cost):
c(140) = 0.6(140)² - 168(140) + 30,389
Calculating c(140):
c(140) = 0.6(19,600) - 23,520 + 30,389
c(140) = 11,760 - 23,520 + 30,389
c(140) = 11,760 + 6,869 = 18,629
Calculating the Minimum Unit Cost
From the calculations above, we find that the minimum unit cost occurs at x = 140 units, with a minimum cost of c(140) = 18,629.
It’s essential for businesses to understand this minimum cost, as it allows them to set production levels that maximize efficiency and minimize costs. This analysis can also assist in pricing strategies and budget allocation.
Impact of Coefficients on Cost Function
The coefficients in the quadratic function significantly influence its shape and position. Here’s a closer look at each coefficient’s impact:
- a (0.6): Determines the width and direction of the parabola. A larger positive value for a would make the parabola narrower, indicating that costs increase rapidly with changes in production levels.
- b (-168): Influences the position of the vertex along the x-axis. A more negative value for b would shift the vertex to the right, affecting the production level at which minimum cost occurs.
- c (30,389): Represents the y-intercept of the function. This value indicates the fixed costs when no units are produced.
Real-World Application of c(x)
Understanding the minimum unit cost derived from the function c(x) = 0.6x² - 168x + 30,389 is vital for businesses in various sectors. For instance:
- Manufacturers can use this analysis to optimize production levels, ensuring they do not exceed costs unnecessarily.
- Startups can determine feasible production scales for their products, helping them to avoid overproduction and excessive costs.
- Financial analysts can evaluate cost efficiency and provide insights to management for strategic decision-making.
Conclusion
In conclusion, the function c(x) = 0.6x² - 168x + 30,389 serves as a valuable tool for determining the minimum unit cost associated with production. By finding the vertex of the function, we established that the minimum unit cost occurs at a production level of 140 units, with a cost of 18,629.
Understanding these concepts is critical for businesses aiming to optimize their operations and manage costs effectively. If you found this article helpful, consider leaving a comment, sharing it with others, or exploring more articles on our site!
Sources
1. "Quadratic Functions." Math is Fun. www.mathsisfun.com
2. "How to Find the Vertex of a Quadratic Function." Purplemath. www.purplemath.com
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