Problem 41 - Algebra 2

Question

A computer manufacturer determines that for each employee the profit for producing \(x\) computers per day is \(P(x)=-0.006 x^{4}+0.15 x^{3}-0.05 x^{2}-1.8 x\) What is the meaning of the roots in this problem?

Step-by-Step Solution

Answer

The roots indicate production levels with zero profit, showing break-even points.

Step 1: Understanding the Profit Function

The profit function given is \( P(x) = -0.006x^4 + 0.15x^3 - 0.05x^2 - 1.8x \), where \( x \) represents the number of computers produced by an employee per day. This is a polynomial equation that models the profit made based on the production quantity.

Step 2: Definition of Roots

The roots of a function are the values of \( x \) where the function equals zero. In this context, the roots \( x \) are the number of computers produced per day where the profit \( P(x) \) is zero.

Step 3: Meaning of the Roots

For this profit function, the roots indicate the production levels at which the profit is zero. This means there is no gain or loss; the company is breaking even. Producing fewer than or more than these amounts could result in either a profit or a loss.

Key Concepts

Roots of a Function

In mathematics, understanding the concept of the "roots of a function" is essential, especially when dealing with polynomial equations like the one in our exercise. Roots are the values of the variable, often denoted as \( x \), that make the entire function equal to zero.
In simpler terms, if you plug a root value back into the function, the result should be zero. For instance, in this problem, the profit function \( P(x) = -0.006x^4 + 0.15x^3 - 0.05x^2 - 1.8x \) has roots that tell us the production levels of computers that lead to zero profit.

• Roots can also be called "zeroes" or "solutions" of the polynomial equation.
• Finding these roots involves solving the equation \( P(x) = 0 \).

In real-world terms, roots can indicate break-even points where financial conditions such as profits neither rise nor fall. This is crucial for businesses as it helps them determine at what production level they aren’t making money or losing money. Thus, understanding roots helps in strategic decision-making.

Profit Function

A profit function is an equation that models how profit changes based on factors such as production level, sales, or costs. In our exercise, the profit function is described by the polynomial \( P(x) = -0.006x^4 + 0.15x^3 - 0.05x^2 - 1.8x \), which relates the number of computers produced each day to the profits.
Let's break it down further:

• The coefficients in the profit function determine the steepness or curvature, affecting how quickly profits increase or decrease.
• Each term in the polynomial, like \( -0.006x^4 \) or \( 0.15x^3 \), contributes differently to the final profit calculation based on \( x \).
• The highest power (degree) of the polynomial, in this case, 4, indicates the complexity and potential turning points of the profit curve.

By analyzing this function, companies can predict expectations of profits as they adjust production levels and make informed decisions.

Polynomial Function Analysis

Polynomial function analysis is an essential tool in understanding complex relationships within various fields. By dissecting a polynomial like our given profit function, we can gain insights into how different factors influence the outcome. Let's delve into its components:

• Degree of the Polynomial: The degree is 4, indicating a quartic equation. This means the graph can have up to 3 turning points, showing potential points of maximum or minimum profit.
• Behavior at the Ends: The leading term \( -0.006x^4 \) suggests that as \( x \) becomes very large or very small, the profit will decrease significantly.
• Identifying Critical Points: By deriving the function and finding where the derivative equals zero, we can locate critical points, which are potential maxima or minima in profit.

Such detailed examination allows businesses to understand and visualize where they might reach peak performance and when they ought to adjust production strategies.