Problem 72 - Algebra and Trigonometry

Question

Estimating Profit An appliance manufacturer estimates that the profit \(y\) (in dollars) generated by producing \(x\) cook tops per month is given by the equation $$ y=10 x+0.5 x^{2}-0.001 x^{3}-5000 $$ (a) Graph the equation. (b) How many cooktops must be produced to begin generating a profit? (c) For what range of values of \(x\) is the company's profit greater than \(\$ 15,000 ?\)

Step-by-Step Solution

Answer

Graph the equation. Begin generating profit at smallest positive \( x \) from \( y = 0 \). Profit > $15,000 for \( x \) satisfying \( y > 20000 \).

Step 1: Understanding the Equation

The profit function given is a cubic equation: \( y = 10x + 0.5x^2 - 0.001x^3 - 5000 \). This function represents the profit \( y \) in dollars for producing \( x \) cooktops.

Step 2: Graphing the Equation

To graph this equation, you need to plot points for different values of \( x \) and then connect them to form a curve. Use technology (like graphing calculators or software) to plot \( y = 10x + 0.5x^2 - 0.001x^3 - 5000 \) and observe its shape.

Step 3: Finding the Break-Even Point

To determine when the company begins to generate profit, set \( y = 0 \) and solve the equation: \( 10x + 0.5x^2 - 0.001x^3 - 5000 = 0 \). Solve this equation using numerical methods or graph analysis to find the smallest positive \( x \) where the function reaches zero, indicating break-even.

Step 4: Solving for Profit Greater than $15,000

To find the number of cooktops that generate a profit greater than $15,000, set \( y > 15000 \). Thus we need to solve \( 10x + 0.5x^2 - 0.001x^3 - 5000 > 15000 \) or equivalently \( 10x + 0.5x^2 - 0.001x^3 > 20000 \). Use numerical methods or graphical analysis to find the range of \( x \) values satisfying this inequality.

Key Concepts

Cubic Function

A cubic function is a polynomial equation of degree three, which means its highest power of the variable is three. In our example, the profit function for the appliance manufacturer is given as:

• \( y = 10x + 0.5x^2 - 0.001x^3 - 5000 \)

Cubic functions have several interesting characteristics:

• They may have up to three real roots, which are the values of \(x\) where the function equals zero.
• The graph often features two turning points, which are locations where the curve changes direction from increasing to decreasing or vice versa.
• Cubic functions can model various real-world problems, including profit maximization scenarios, because they can represent more complex relationships compared to linear or quadratic functions.

In our equation, each term adds to the complexity and dynamic nature of the curve formed when the function is plotted. The \(-5000\) represents a fixed cost or expense, a constant that shifts the profit up or down. The term \(-0.001x^3\) introduces a cubic nature to the function, affecting the curve's slope at higher productions.

Graphing Equations

Graphing equations like our profit function involves plotting points for different values of \(x\) and observing the resulting shape. In this particular example, the equation to be graphed is:

• \( y = 10x + 0.5x^2 - 0.001x^3 - 5000 \)

Steps for graphing:

• Choose a range of \(x\) values, preferably around the expected break-even points and peak areas. Automated tools like graphing calculators or software make this process more efficient.
• Calculate the corresponding \(y\) values for these \(x\) points.
• Plot these points on a graph and connect them to reveal the cubic curve, showing oscillations and turning points.

In our case, the graph helps identify where profit becomes positive (break-even point) and the production levels yielding the highest profit, crucial for making informed business decisions. Observing the curve's increase and decrease, we determine economically viable production ranges where profits exceed given thresholds.

Break-Even Analysis

Break-even analysis is essential for determining the point at which a business covers all its costs, resulting in zero profit or loss. For the given cubic profit function, determining the break-even point involves:

• Setting the profit equation to zero: \(10x + 0.5x^2 - 0.001x^3 - 5000 = 0\).
• Finding the smallest positive \(x\) where the equation returns zero, indicating the minimum production needed to start making a profit.

Numerical or graphical methods can be used effectively to solve this. By graphing the profit equation, you can visually inspect where the curve crosses the \(x\)-axis. At this intersection, the profit becomes zero, marking the break-even point.
Beyond this break-even analysis, if we want to assess where the profit exceeds $15,000, we adjust our equation to reflect:

• \(10x + 0.5x^2 - 0.001x^3 > 20000 \).

Similar methods are used to identify \(x\) values within this range, providing insights into optimal production levels for desired profitability.