Problem 31 - Algebra A Combined Function

Question

Solve. The cost \(C\) in dollars of manufacturing \(x\) bicycles at Holladay's Production Plant is given by the function \(C(x)=2 x^{2}-800 x+92,000\) a. Find the number of bicycles that must be manufactured to minimize the cost. b. Find the minimum cost.

Step-by-Step Solution

Answer

a. 200 bicycles, b. $12,000

Step 1: Identify the type of function

The function given is a quadratic function of the form \(C(x) = ax^2 + bx + c\), where \(a = 2\), \(b = -800\), and \(c = 92,000\). Quadratic functions have parabolic graphs which have either a minimum or maximum point, determined by the coefficient \(a\). Since \(a = 2 > 0\), the parabola opens upwards, indicating there is a minimum point.

Step 2: Find the x-value of the vertex

The x-value of the vertex of a parabola given by the function \(ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\). Substitute \(a = 2\) and \(b = -800\) into the formula: \[ x = -\frac{-800}{2 \times 2} = \frac{800}{4} = 200. \]The number of bicycles that must be manufactured to minimize the cost is 200.

Step 3: Calculate the minimum cost

Substitute \(x = 200\) back into the cost function \(C(x) = 2x^2 - 800x + 92,000\) to find the minimum cost:\[C(200) = 2(200)^2 - 800(200) + 92,000.\]Calculate each term:\(2(200)^2 = 80,000\)\(-800(200) = -160,000\)Now calculate the full expression:\[ C(200) = 80,000 - 160,000 + 92,000 = 12,000. \] The minimum cost is $12,000.

Key Concepts

Parabolas

Parabolas are a type of curve on a graph that are part of the family of quadratic functions. These curves resemble the shape of a U or an upside-down U. The overall shape of a parabola is determined by its leading coefficient, which is the number in front of the squared term in the quadratic equation.
For example, consider the quadratic function:

• When the leading coefficient (\( a \) ) is positive (greater than zero), the parabola opens upwards, creating the shape of a U.
• Conversely, when (\( a \) ) is negative (less than zero), the parabola opens downwards, resembling an upside-down U.

The special features of parabolas are their vertices and axes of symmetry. The "vertex" is the point where the parabola changes direction; it is either the highest or lowest point on the curve. Additionally, parabolas are symmetrical around a vertical line that goes through the vertex. Hence, understanding parabolas is vital in analyzing various mathematical models, including those related to cost functions in manufacturing.

Vertex Calculation

Calculating the vertex is a key step in utilizing quadratic functions to solve practical problems. In the context of cost functions, finding the vertex helps us determine the minimum cost or resource use at a manufacturing plant, like Holladay's Production Plant.
To find the vertex of a quadratic function given by:\( ax^2 + bx + c \), we use the formula for the x-value of the vertex: \[ x = -\frac{b}{2a} \].
For instance, in the function included in the exercise: \( C(x) = 2x^2 - 800x + 92,000 \):

• Substitute (\( a = 2 \) and \( b = -800 \)) into the vertex formula: \[ x = -\frac{-800}{2 \times 2} = \frac{800}{4} = 200 \].
• This means that the minimum cost occurs when 200 bicycles are manufactured.

This calculated x-value of 200 not only shows us the vertex location but also indicates how to use the application of quadratic functions in real-world manufacturing issues.

Minimum Cost in Manufacturing

In many manufacturing settings, understanding how to determine the minimum cost is crucial for efficiency and savings. By using the vertex calculation derived from quadratic functions, businesses can optimize the number of goods produced to minimize costs. In the given exercise, we already determined that the minimum cost corresponds to the production of 200 bicycles.
To find the actual minimum cost, we need to substitute this x-value into the original cost function: \[ C(x) = 2x^2 - 800x + 92,000 \].
Using 200 in place of x results in:

• First, compute \( 2(200)^2 = 80,000 \).
• Next, calculate \(-800(200) = -160,000\).
• Finally, evaluate the function: \[ C(200) = 80,000 - 160,000 + 92,000 = 12,000 \].

The minimum cost to manufacture 200 bicycles is $12,000. This approach not only highlights the practical use of mathematical concepts like quadratic functions but also outlines a clear pathway to cost efficiency.