Problem 43 - Introductory Algebra for College Students

Question

The personnel manager of a roller skate company knows that the company's weekly revenue, \(R,\) in thousands of dollars, can be modeled by the formula $$R=-2 x^{2}+36 x$$ where \(x\) is the price of a pair of skates, in dollars. A job applicant promises the personnel manager an advertising campaign guaranteed to generate 200,000 dollar in weekly revenue. Substitute 200 for \(R\) in the given formula and solve the equation. Are the solutions real numbers? Explain why the applicant will or will not be hired in the advertising department.

Step-by-Step Solution

Answer

The price for the roller skates cannot be determined to achieve the promised 200,000 dollars weekly revenue based on the company’s current revenue model. The solutions to the equation are imaginary numbers, hence there's no viable price in real dollars that would generate the proposed revenue. The decision would be not to hire the applicant.

Step 1: Substitute \(R\) with 200

First, the given revenue \(R = -2x^2 + 36x\) needs to be equal to 200. So we form the equation as -2x^2 + 36x = 200.

Step 2: Arrange Equation in Standard Quadratic Form

In order to solve this, we need to rearrange the equation in the standard form of a quadratic equation (\(ax^2 + bx + c = 0\)). So the equation will be -2x^2 + 36x - 200 = 0.

Step 3: Solving the Quadratic Equation

Let's solve the quadratic equation: Step 1: find the discriminant. The discriminant \(D\) is given by \(D = b^2 - 4ac\), where a, b and c are the coefficients of the quadratic equation. Here, a = -2, b = 36 and c = -200. Substituting these values gives \(D = (36)^2 - 4*(-2)*(-200) = 1296 - 1600 = -304.\)Step 2: find the roots of the equation. The roots of a quadratic equation are given by \(x = [-b +- sqrt(D)] / 2a\). Because our discriminant D is negative, the roots will be imaginary, not real numbers. Thus no real value of \(x\) satisfies the equation.

Step 4: Summary and Decision for Hiring

The roots are imaginary, meaning there are no real solutions. Therefore, it's not possible to generate 200,000 dollars of revenue given the current formula for revenue. This suggests that the applicant's claim is unfounded, and they shouldn't be hired.

Key Concepts

Discriminant in Quadratic Equations

In solving quadratic equations, the first step often involves calculating the discriminant. The discriminant is a key part of the quadratic formula \(x = \frac{{-b \pm \sqrt{D}}}{{2a}}\), where \(D = b^2 - 4ac\). It provides important information about the nature of the roots of a quadratic equation.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root, also known as a double root.
• If the discriminant is negative, the roots are not real numbers; instead, they are imaginary numbers.

When the discriminant is negative, like \(D = -304\) in our example, it indicates that our equation has imaginary roots. This happens because you cannot take the square root of a negative number using real numbers. Hence, no real solution exists for such an equation.

Imaginary Numbers

Imaginary numbers come into play when we're dealing with the square root of negative numbers. Typically, the square root of negative one is defined as the imaginary unit \(i\). For example, \(\sqrt{-1} = i\).
When a quadratic equation has a negative discriminant, its roots will include \(i\). This means the solutions are complex numbers of the form \(a \pm bi\), where \(a\) and \(b\) are real numbers. In the example provided, the negative discriminant \(D = -304\) results in solutions such as \(x = \frac{{-b \pm \sqrt{-304}}}{{2a}}\), which simplifies to complex numbers.
Imaginary numbers are very useful in various fields like electrical engineering and physics, but in this revenue modeling case, having imaginary solutions implies the applicant's revenue promise isn't feasible.

Revenue Modeling

Revenue modeling is an approach used by businesses to estimate potential income. The model's mathematical representation helps predict how changes in input—like product pricing—affect revenue. The quadratic function \(R = -2x^2 + 36x\) models the relationship between the skate price and revenue.
This equation's purpose is to understand the behavior of revenue with respect to different pricing strategies.

• The coefficient \(-2\) indicates a parabolic decrease in revenue after a certain price point, suggesting there's an optimal price for maximum revenue.
• This reflects real-world business scenarios where very high or very low prices can lead to decreased revenue.

In this task, setting the revenue to generate $200,000 leads to an equation with no real solutions, showing that reaching that specific revenue with current pricing isn't possible. Thus, despite the promise of the job applicant, the revenue model suggests their claim is unsubstantiated and unrealistic.