Optimizing Marginal Cost: An In-Depth Analysis of the Average Cost Function
When examining the cost structure of a business, one of the key factors to focus on is the cost optimization. This involves understanding and utilizing the relationship between average cost and marginal cost. In this article, we will delve into the process of finding the x units at which the marginal cost is at its minimum using the given equation: Average Cost (1/3x^2) - 18x 160. We will explore both graphical and algebraic methods to achieve this goal.
Understanding the Relationship Between Average Cost and Marginal Cost
Before we dive into the problem, it's essential to understand the definitions of average cost (AC) and marginal cost (MC). The average cost is the total cost divided by the number of units produced, representing the cost per unit. Mathematically, this can be written as:
AC (Total Cost) / (Number of Units)
The marginal cost is the change in total cost when one additional unit is produced, reflecting the additional cost of producing one more unit. Thus, in this case, the marginal cost is derived as the derivative of the total cost with respect to the number of units produced.
Deriving the Marginal Cost Function
Given the average cost function:
AC (1/3)x^2 - 18x 160
The marginal cost (MC) is the derivative of the average cost function with respect to (x). To find this, we will differentiate the given equation:
MC d(AC)/dx d(1/3x^2 - 18x 160)/dx
Employing basic differentiation rules:
MC (2/3)x - 18
To find the minimum marginal cost, we need to find the critical points by setting the derivative of MC equal to zero:
(2/3)x - 18 0
Solving for (x):
(2/3)x 18
x 18 * (3/2)
x 27
Graphical Analysis of the Marginal Cost Function
To provide a visual representation, let's graph the marginal cost function, (MC (2/3)x - 18).
Graphically, the marginal cost function is a straight line with a positive slope. We can plot the function on a graph using the y-intercept -18 and the slope of 2/3.
The minimum point on this line corresponds to the vertex of the parabola, which in this case is where the linear function intersects the x-axis.
Using a graphing calculator or plotting points by hand, we can confirm that the x-value at which the marginal cost is minimized is indeed (x 27).
Conclusion
In conclusion, by analyzing the given average cost function (AC (1/3)x^2 - 18x 160), we have determined that the marginal cost is minimized at (x 27) units. This optimization is crucial for businesses looking to minimize costs and maximize profit. Both the algebraic and graphical methods have consistently provided the same result, confirming the accuracy of our findings.
Understanding and applying these concepts can significantly help businesses in making informed decisions regarding production levels and cost management. Whether you're a business owner, a marketing professional, or a student studying economics, this knowledge can prove invaluable.