Problem 58 - 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 $$ y=10 x+0.5 x^{2}-0.001 x^{3}-5000 $$ where \(0 \leq x \leq 450\) (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

(a) Graph the cubic equation for a visual understanding. (b) Calculate the breakeven point by solving \(10x + 0.5x^2 - 0.001x^3 = 5000\). (c) Identify \(x\) where \(y > 15000\).

Step 1: Graph the Equation

To graph the equation, substitute different values of \(x\) within the given range \(0 \leq x \leq 450\) into the equation \(y = 10x + 0.5x^2 - 0.001x^3 - 5000\). Plot the corresponding \(y\) values against \(x\) on a coordinate plane to visualize the behavior of the profit function. The graph will be a cubic parabola showing how profit changes with varying production levels.

Step 2: Determine Breakeven Point

To find the minimum production level \(x\) required to generate a profit, set the profit equation \(y = 10x + 0.5x^2 - 0.001x^3 - 5000\) to zero and solve for \(x\). Use numerical or graphical methods to solve: \(10x + 0.5x^2 - 0.001x^3 - 5000 = 0\). The smallest positive \(x\) where \(y\) becomes positive represents the breakeven point where profit begins.

Step 3: Find Range for Profit Greater Than $15,000

To determine the values of \(x\) where \(y > 15000\), rearrange the equation to \(10x + 0.5x^2 - 0.001x^3 - 5000 > 15000\), simplify to \(10x + 0.5x^2 - 0.001x^3 - 20000 > 0\), and solve for \(x\). This requires finding the roots or using a graph to locate the interval \(x\) where the inequality holds true.

Key Concepts

Cubic Functions

Cubic functions are mathematical expressions where the highest degree of a variable is three, typically written as \( ax^3 + bx^2 + cx + d \). These functions are part of polynomial equations and have unique properties:

• They can have one or more turning points where the graph changes direction.
• They can intersect the x-axis up to three times, meaning they can have up to three real roots.
• Their graph can be asymmetric, with one end going to positive infinity and the other to negative infinity.

Understanding cubic functions is crucial for solving complex real-world problems, such as estimating profit. The cubic function in our problem, \( y = 10x + 0.5x^2 - 0.001x^3 - 5000 \), helps in identifying how profit varies with the number of products produced. It is tailored to show diminishing returns, as seen by the negative cubic term \(-0.001x^3\), which reduces profit as production goes beyond certain levels.

Graphing Equations

Graphing equations is an essential skill in visualizing mathematical relationships. For the given problem, we graph the profit function \( y = 10x + 0.5x^2 - 0.001x^3 - 5000 \) to understand how profit changes with the production number. Here’s a simple way to graph equations:

• Select a range for your variable \( x \), in this case from 0 to 450.
• Substitute various values of \( x \) into the equation to compute corresponding \( y \) values.
• Plot these \( (x, y) \) pairs on a coordinate plane.
• Look for the overall shape and critical points such as maxima, minima, and intercepts.

By graphing the equation, you can visually determine the behavior of profit. For instance, this graph might reveal a peak profit level and help approximate where profits start or cease.

Breakeven Analysis

Breakeven analysis is used to determine when a business will begin to generate profit. This is achieved by setting the profit function equation to zero to solve for \( x \), the quantity that needs to be produced to reach break-even.

• For the equation \( y = 10x + 0.5x^2 - 0.001x^3 - 5000 \), the calculation involves solving \(10x + 0.5x^2 - 0.001x^3 - 5000 = 0\).
• One way to solve this is to use numerical methods, such as a graphing calculator, to identify the smallest positive root.
• Graphical solutions can also be effective, allowing you to find the breakeven point where the graph crosses the x-axis.

Once the breakeven point is determined, it's easier to make informed decisions on production levels and forecast financial outcomes.

Inequality Solutions

Solving inequalities involves finding values that satisfy a condition expressed as \( >, <, \geq, \) or \( \leq \). In the context of our exercise, we need to find when the company's profit is greater than $15,000.To find the values of \( x \) where \( y > 15000 \):

• First, set the inequality \( 10x + 0.5x^2 - 0.001x^3 - 20000 > 0 \).
• Use algebraic manipulation or graphing to identify intervals where this holds true.
• Graphical methods are particularly useful in visualizing these regions; look for segments above the threshold on the profit curve.

By determining inequality solutions, businesses can make strategic decisions on optimal production to maximize profit.