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Chapter 2: Problem 18

Suppose \(C(x)=x^{2}-10 x+27\) represents the costs, in hundreds, to produce \(x\) thousand pens. How many pens should be produced to minimize the cost? What is this minimum cost?

Short Answer

Expert verified
Produce 5,000 pens for a minimum cost of $200.

Step by step solution

01

Identify the Quadratic Function

The cost function is given as a quadratic function: \(C(x) = x^2 - 10x + 27\). This is a standard quadratic function in the form \(ax^2 + bx + c\).
02

Find the Vertex of the Parabola

The quadratic function \(C(x) = ax^2 + bx + c\) will have its minimum value at the vertex, since the parabola is opening upwards (the coefficient of \(x^2\) is positive). The x-coordinate of the vertex can be found using the formula \(x = -\frac{b}{2a}\).
03

Calculate the x-coordinate for the Minimum Cost

Here, \(a = 1\) and \(b = -10\). Substitute these into the formula: \(x = -\frac{-10}{2 \cdot 1} = \frac{10}{2} = 5\). Therefore, 5 thousand pens need to be produced to minimize the cost.
04

Substitute Back to Find the Minimum Cost

Now substitute \(x=5\) back into the cost function to find the minimum cost. \[C(5) = (5)^2 - 10(5) + 27 = 25 - 50 + 27 = 2\].
05

Interpret the Result

Since the function represents cost in hundreds, the minimum cost is \(2\times 100 = 200\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
Quadratic functions are mathematical expressions of the form \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. They form parabolas when graphed on a coordinate plane. In these functions, the term \( ax^2 \) gives the function its characteristic u-shape, known as a parabola.
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), it opens downwards.
Understanding the coefficients is crucial:
  • \(a\) influences the direction and width of the parabola.
  • \(b\) affects the position of the parabola along the x-axis.
  • \(c\) determines the y-intercept, the point where the parabola crosses the y-axis.
Recognizing these elements helps in analyzing the behavior of the quadratic function.
Vertex of a Parabola
The vertex of a parabola is a crucial point that represents the turning point of the quadratic function, whether it is a minimum or maximum point.It can be found using the formula \( x = -\frac{b}{2a} \). This determines the x-coordinate, and substituting back into the equation gives the y-coordinate.
  • For \( C(x) = x^2 - 10x + 27 \), the vertex formula gives us \( x = \frac{10}{2} = 5 \).
  • Substituting \( x=5 \) back into \( C(x) \) provides the y-coordinate: \[ C(5) = 25 - 50 + 27 = 2 \].
This point \((5, 2)\) is the vertex of the parabola and represents the minimum cost in this scenario.Understanding the vertex is key to interpreting the quadratic's graph and its real-world applications.
Cost Minimization
Cost minimization is an essential concept in various fields like economics, business, and operations.In the context of quadratic functions, the goal is to minimize the cost associated with production, represented by the parabola's vertex. For the quadratic \( C(x) = x^2 - 10x + 27 \), minimizing cost means finding the minimum point of its parabola.
  • Calculating \( x = 5 \) shows the optimal production at which costs are minimized.
  • Substituting this \( x \) into the original function gives a minimum cost of 200 dollars (since \( C(x) \) is in hundreds).
Focusing on cost minimization allows businesses to optimize resources and make informed decisions about production levels.This ensures efficiency and can save significant resources.
Graphing Parabolas
Graphing parabolas helps visualize the behavior of quadratic functions. By plotting key points, like the vertex, the direction in which the parabola opens, and intercepts, one can gain insights into the function's properties.For \( C(x) = x^2 - 10x + 27 \):
  • The parabola opens upwards because \( a = 1 \) is positive.
  • The vertex found at \((5, 2)\) indicates the parabola鈥檚 lowest point.
  • The y-intercept at \((0, 27)\) shows where it crosses the y-axis.
Drawing a smooth curve through these points completes the graph.Graphing quadratics is not just a mathematical exercise; it provides an intuitive understanding of cost dynamics and optimization in practical situations. Using graphs, one can easily identify impact points and trends.

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