Problem 40 - Algebra 2

Question

A computer manufacturer determines that for each employee the profit for producing \(x\) computers per day is \(P(x)=-0.006 x^{4}+0.15 x^{3}-0.05 x^{2}-1.8 x\) Approximate all real zeros to the nearest tenth by graphing the function using a graphing calculator.

Step-by-Step Solution

Answer

The approximate real zeros of the function are at \( x \\approx 0, 12.7 \).

Step 1: Understanding the Task

We need to find the values of \( x \) (the number of computers produced per day) where the profit \( P(x) \) is zero. This means we must find the zeros or roots of the polynomial equation \( P(x) = -0.006x^4 + 0.15x^3 - 0.05x^2 - 1.8x \).

Step 2: Graphing the Function

I will input the function \( P(x) = -0.006x^4 + 0.15x^3 - 0.05x^2 - 1.8x \) into a graphing calculator to plot the graph. The graph will help visualize where the function intersects the x-axis, which corresponds to the zeros of the function.

Step 3: Identifying the Zeros from the Graph

After plotting the graph, I will observe the points where the curve crosses the x-axis. These intersections represent the real zeros of the polynomial function. I will identify the \( x \)-values at these intersection points.

Step 4: Approximating and Recording the Zeros

Using the graph, I will approximate each zero to the nearest tenth. I will then list these values as the solution to the problem.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can assist in visualizing complex equations and functions. When dealing with polynomial equations, visual representation is invaluable for identifying key characteristics, such as zeros or x-intercepts.

• First, input the polynomial equation carefully to avoid errors.
• Then, examine the graph to find where it crosses the x-axis.
• Adjust the calculator's window settings to ensure the graph is clear and detailed.

By analyzing these intersection points, you can efficiently approximate the real zeros of the equation, which can otherwise be a complex task when solved algebraically.

Profit Function

The profit function represents a mathematical model that determines profit based on the number of units produced. In this case, the manufacturer uses the profit function:
\[ P(x) = -0.006x^4 + 0.15x^3 - 0.05x^2 - 1.8x \] This polynomial shows how producing a certain number of computers affects profit. Notice

• The coefficients and powers of x influence various aspects of the profit structure.
• The terms include a mix of negative and positive coefficients, showing that profit changes non-linearly.

Understanding these dynamics helps businesses maximize profits by adjusting their production levels accordingly.

Real Zeros

Real zeros of a polynomial function represent the values of \( x \) where the output, or profit in this instance, equals zero. Finding these zeros is crucial because:

• They mark points where the profit is neutral, offering a breakeven analysis.
• Each zero corresponds to a specific production level where no profit is made.

In practical terms, identifying real zeros can help manufacturers make strategic decisions about production volume in relation to profitability.

Polynomial Equation

A polynomial equation is an expression consisting of variables and coefficients, related by operations of addition, subtraction, multiplication, and non-negative integer exponents. The given polynomial equation can be broken down as:

• A degree of 4, indicated by the highest exponent on x, which suggests complexity in the behavior of the graph.
• The solution to the equation requires determining where the polynomial equals zero, which involves finding its roots or zeros.

Polynomial equations often model real-world scenarios effectively, capturing various phenomena through their terms and coefficients. Understanding how to manipulate and interpret these equations is essential for problem-solving in diverse scientific and engineering fields.