Problem 53 - College Algebra

Question

Suppose that an equation is given \(p=-2 x^{2}+280 x-1000,\) where \(x\) represents the number of items sold at an auction and \(p\) is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using \(2^{\text { nd }}\) CALC maximum. To obtain a good window for the curve, set \(x[0,200]\) and \(y[0,10000].\)

Step-by-Step Solution

Answer

100 items will yield the maximum profit.

Step 1: Understand the Equation

We are given the profit equation: \( p = -2x^2 + 280x - 1000 \). This represents a quadratic equation, where \( x \) is the number of items sold and \( p \) is the profit.

Step 2: Define the Graphing Window

To graph the equation, set the window for \( x \) as \([0, 200]\) and for \( y \) as \([0, 10000]\). This helps in visualizing the curve properly and finding the maximum point accurately.

Step 3: Use Graphing Utility

Input the equation \( p = -2x^2 + 280x - 1000 \) into your graphing calculator or graphing utility. This will display the parabola representing the profit based on the number of items sold.

Step 4: Identify the Vertex

Since the parabola opens downward (the coefficient of \( x^2 \) is negative), the maximum profit occurs at the vertex of the parabola.

Step 5: Use Calculator Function to Find Maximum

Utilize the calculator's "2nd CALC" feature and select 'maximum' to find the vertex of the parabola. You'll select a point just left of the vertex, a point just right of the vertex, and then guess a point near where you believe the maximum is.

Step 6: Read the Maximum Point

The calculator will display the coordinates of the maximum point. The \( x \)-coordinate represents the number of items sold that results in maximum profit.

Key Concepts

Profit Maximization

Profit maximization is a key concept in business, referring to the point at which a company sees the highest possible profit from their operations. In the case of quadratic equations like \[p = -2x^2 + 280x - 1000,\]the goal is to find the highest value of \( p \) (profit), which will occur at a specific \( x \) (number of items sold). This is critical because selling either too few or too many items can reduce overall profit.
In real-world applications, companies use such models to determine optimal sales volumes. By identifying the maximum profit point, businesses can adjust their strategies on production, pricing, and marketing.
To locate this maximum profit in quadratic equations, the vertex of the parabola plays a vital role. This point, where the graph turns, provides us with the information needed to achieve the most desirable financial outcome.

Graphing Utility

A graphing utility is a tool, often software-based, used to plot and analyze mathematical equations visually. This tool is linearly invaluable for solving quadratic equations, as it provides a clear, visual representation of the relationship between variables.
To effectively use a graphing utility for a quadratic equation, you need to input the equation correctly, here it's\[-2x^2 + 280x - 1000.\]
Next, you establish the graph window, selecting reasonable limits for both axes; in our example, these were \([0, 200]\) for \(x\) and \([0, 10000]\) for \(y\).
Once set, the graphing utility will draw the parabola, allowing you to see the curve and identify key features like the vertex, intercepts, and axis of symmetry. This visual cue is essential to understanding how variations in \(x\) affect the profit \(p\), helping users make informed decisions based on the graphical data displayed.

Parabola Vertex

The vertex of a parabola is a crucial element in understanding quadratic functions, especially in the context of maximization or minimization problems. For the quadratic equation \[p = -2x^2 + 280x - 1000,\]the parabola opens downward because the \(x^2\) term coefficient is negative.
The vertex of this parabola signifies the maximum profit point. To find the vertex in any quadratic equation of the form \(ax^2 + bx + c\), you'll use the formula:\[x = \frac{-b}{2a}.\]
Applying it here gives us \[x = \frac{-280}{2(-2)} = 70.\]
This calculation shows that the business will maximize its profit by selling 70 items. Understanding the vertex is not just about finding where the curve peaks but also about interpreting its implications in a real-world business context.

Graphing Calculator

A graphing calculator is an electronic device capable of plotting graphs, solving simultaneous equations, and performing other tasks with variables. It serves as a powerful tool for students and professionals dealing with complex mathematical equations.
In solving the equation \[p = -2x^2 + 280x - 1000,\]you use this calculator to graph the equation and visually locate the vertex of the parabola — where maximum profit occurs. By accessing the '2nd CALC' function, you can precisely determine the maximum point by selecting points around the vertex.
This feature is particularly useful in educational settings where visual learning aids comprehension. It allows students to engage interactively with mathematical concepts, better understand the behavior of quadratic equations, and appreciate the application of mathematics in practical profit-maximization scenarios.