Problem 39 - Intermediate Algebra

Question

Use quadratic functions. Suppose that the cost function for a particular item is given by the equation \(C(x)=2 x^{2}-320 x+12,920\), where \(x\) represents the number of items. How many items should be produced to minimize the cost?

Step-by-Step Solution

Answer

80 items should be produced.

Step 1: Identify the form of the equation

The given cost function is a quadratic function written as \( C(x) = ax^2 + bx + c \) where \( a = 2 \), \( b = -320 \), and \( c = 12,920 \). This is in the standard form of a quadratic equation.

Step 2: Determine the vertex formula

To find the value of \( x \) that minimizes the cost, we need to find the vertex of the parabola represented by the quadratic function. The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \).

Step 3: Plug in the values

Substitute \( b = -320 \) and \( a = 2 \) into the vertex formula: \[x = -\frac{-320}{2 \times 2} = \frac{320}{4} = 80\]

Step 4: Interpret the result

The vertex of the quadratic function indicates the point of minimum cost because the parabola opens upwards (since \( a > 0 \)). Therefore, the minimum cost occurs when \( x = 80 \).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that represents either the minimum or maximum value of a quadratic function. This hinges on the direction the parabola opens, which is determined by the sign of the coefficient 'a' in the quadratic equation of the form \[ ax^2 + bx + c. \]

• If a is positive, the parabola opens upwards, and the vertex is the minimum point.
• If a is negative, the parabola opens downwards, and the vertex is the maximum point.

To find the vertex, we use a specific formula. The x-coordinate of the vertex is calculated with the formula \[ x = -\frac{b}{2a}, \] where b and a are the coefficients from the quadratic equation. After finding the x-coordinate, you can substitute it back into the quadratic equation to find the corresponding y-coordinate, giving you the exact point of the vertex.
The vertex assists in understanding the function's behavior and plays a major role in applications such as physics and economics by defining optimal points.

Minimizing Quadratic Functions

Quadratic functions can model various real-world scenarios, including situations where optimization is required. With a cost function, our objective is usually to minimize or maximize a certain quantity. For minimizing quadratic functions, the vertex is your main ally.
Whenever your quadratic function has a positive 'a' value, the vertex symbolically sits at the bottom of the 'U' shape of a parabola, representing a local minimum. In such cases, finding the vertex will ensure that you've identified the minimum point of the function, thus optimizing your parameters.
By using the vertex formula \[ x = -\frac{b}{2a}, \]you can quickly find this optimal point. As in the original exercise where known coefficients were used, substituting these gives you the ideal number of items to minimize cost.

• This exploration not only provides the essential x-value but also a clearer understanding of the efficient allocation of resources.

Cost Function Analysis

Cost function analysis involves understanding and interpreting a function to make economical decisions. When expressed as a quadratic function, it typically takes the form \[ C(x) = ax^2 + bx + c, \] where C(x) represents the total cost associated with producing x units of a product. The coefficients dictate the cost structure:

• The coefficient a impacts the rate at which costs increase.
• The coefficient b represents the cost per unit.
• c significantly influences total fixed costs unrelated to production volume.

Analyzing these functions allows businesses to identify optimal production levels (as indicated by the vertex). This level of production minimizes cost, enhances efficiency, and maximizes profitability.
In practical terms, this suggests producing 80 items (from our exercise context) minimizes costs, providing a strategic insight for manufacturing decisions. By appreciating these nuances, firms can make well-rounded, profit-boosting decisions, just by interpreting the quadratic nature of their cost functions.