Problem 58 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The total revenue realized by the Apollo Company from the sale of \(x\) PDAs is given by \(R(x)=-0.1 x^{2}+500 x\) dollars. Factor the expression on the right- hand side of this equation.

Step-by-Step Solution

Answer

The factored expression of the total revenue function, \(R(x) = -0.1x^2 + 500x\), is \(R(x) = x(-0.1x + 500)\).

Step 1: Factor out common factor

\(R(x) = -0.1x^2 + 500x = x(-0.1x + 500)\)

Step 2: Alternative form

This can also be written as \(R(x) = -0.1x(x - 5000)\).

Key Concepts

Quadratic Revenue Models

Understanding quadratic revenue models is essential for students studying economics, business, or any field that involves profitability analysis. A quadratic revenue model is represented by a quadratic equation, which is an equation of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. In the context of the Apollo Company's problem, the total revenue \( R(x) \) is modeled by the equation \( R(x) = -0.1x^2 + 500x \).

The quadratic nature of the revenue model reflects the reality that revenue does not increase indefinitely as sales (\( x \)) increase. Initially, the revenue might rise with every PDA sale, but after a certain point, the increase in production could lead to a smaller profit margin, or market saturation might reduce demand. This real-world dynamic is captured perfectly in the downward-facing parabola of the quadratic equation, where the coefficient of the \( x^2 \) term is negative, indicating that the revenue will eventually decrease after reaching a maximum point.

Common Factor Extraction

Common factor extraction is a method used in algebra to simplify expressions and solve equations more efficiently. It involves identifying and removing a common factor from each term in a polynomial. In the context of our Apollo Company problem, we are given a quadratic expression \( -0.1x^2 + 500x \). By examining the terms, we notice that both terms include the variable \( x \) as a factor.

We can then 'extract' this common factor, simplifying the expression. Here, the term \( x \) is present in both \( -0.1x^2 \) and \( 500x \) terms, so it is factored out, as shown in the step-by-step solution. This simplification not only makes the equation more manageable but also prepares us for further steps in solving the equation, which may include finding the roots of the equation or analyzing its properties graphically.

Polynomial Factoring

Polynomial factoring is a crucial skill in algebra that allows students to break down complex expressions into simpler, more usable components. The process of factoring can simplify computations, provide insights into the properties of the original equation, and help solve for the roots of the equation. In the quadratic revenue model provided, the polynomial expression \( R(x) = -0.1x^2 + 500x \) is factored by first extracting the common factor of \( x \).

The result is \( R(x) = x(-0.1x + 500) \), which is a factored form where \( x \) is the common factor, and the expression in the parenthesis is a binomial. This process reduces the polynomial from a quadratic to a linear expression, multiplied by \( x \) which itself is a solution to the equation (specifically, the x-intercept). Factoring is often a prerequisite for further solving polynomials, which could also lead to finding the vertex of the parabola that represents the revenue model graphically and determining the maximum revenue possible.