Problem 44 - Algebra for College Students

Question

Suppose that the cost function for the production of 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

Produce 80 items to minimize the cost.

Step 1: Identify the Cost Function

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

Step 2: Formula for Vertex of a Parabola

The cost function is quadratic, representing a parabola that opens upwards because the coefficient of \(x^2\), \(a = 2\), is positive. The minimum point of the parabola, or the vertex, will give the number of items \(x\) that minimizes the cost. The \(x\)-coordinate of the vertex is given by the formula \( x = -\frac{b}{2a} \).

Step 3: Substitute Values into Vertex Formula

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

Step 4: Solve for x

Calculate the value of \(x\):\[ x = \frac{320}{4} = 80 \]. Thus, producing 80 items will minimize the cost.

Key Concepts

Cost Function

A cost function is a mathematical representation that describes the total cost of producing a certain number of goods. In our example, the cost function is expressed as a quadratic equation: \( C(x) = 2x^2 - 320x + 12,920 \).
Here, \( C(x) \) is the total cost, and \( x \) represents the number of items produced.

• The first term, \( 2x^2 \), is the quadratic term, which indicates how costs increase with the square of production amount.
• The second term, \( -320x \), is the linear term, affecting cost proportionally to the number of items.
• The constant term, \( 12,920 \), represents a fixed cost regardless of production levels.

Understanding this function is crucial for determining how changes in production affect overall costs.

Parabola Vertex

In a quadratic equation like our cost function, the graph will form a shape called a parabola. The vertex of the parabola is a critical point, providing either a maximum or minimum value of this function, depending on its orientation.
Since our parabola opens upwards (because the coefficient \( a = 2 \) is positive), the vertex represents the minimum cost point.
To locate the vertex in terms of \( x \), we use the formula for finding the \( x \)-coordinate of the vertex:

• \( x = -\frac{b}{2a} \)

This formula helps determine the precise number of items needed to achieve minimum costs.

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two, often written in the standard form \( ax^2 + bx + c = 0 \).
It is characterized by:

• Two changes (increases or decreases) in the curve's slope.
• A graph shape that is always a parabola.

In quadratic optimization problems, like ours, solving this equation helps determine key points like the vertex that represent optimal solutions or cost efficiencies.
These optimal points are invaluable in making production decisions based on minimizing costs.

Minimum Cost Calculation

Calculating the minimum cost involves determining the number of items that lead to the lowest possible value of the cost function. This is achieved by:

• Finding the vertex of the parabola, which marks the minimum cost point.
• Utilizing the vertex formula \( x = -\frac{b}{2a} \), substituting the specific coefficients to find \( x \).
• Solving for \( x \) gives the number of items—80 in this scenario—that should be produced to achieve minimal cost.

This straightforward computation helps businesses decide on optimal production levels to reduce costs and increase efficiency, highlighting the practical power of quadratic optimization in real-world applications.