Problem 81 - Algebra and Trigonometry

Question

Market Research A market analyst working for a small- appliance manufacturer finds that if the firm produces and sells \(x\) blenders annually, the total profit (in dollars) is $$P(x)=8 x+0.3 x^{2}-0.0013 x^{3}-372$$ Graph the function \(P\) in an appropriate viewing rectangle and use the graph to answer the following questions. (a) When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, \(P(10)=-263.3\) so the firm loses \(\$ 263.30\) if it produces and sells only 10 blenders.) How many blenders must the firm produce to break even? (b) Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?

Step-by-Step Solution

Answer

(a) Break-even at approximately 1075 blenders. (b) Largest profit at about 800 blenders.

Step 1: Understanding the Profit Function

The profit function is given as \( P(x) = 8x + 0.3x^2 - 0.0013x^3 - 372 \). This is a cubic polynomial that represents the total profit based on the number of blenders \(x\) sold.

Step 2: Setting the Break-even Point

To break even, the profit \( P(x) \) must be zero. We set the equation to zero: \\[ 8x + 0.3x^2 - 0.0013x^3 - 372 = 0 \] \We will solve this equation numerically to find when the profit equals zero.

Step 3: Solving for Break-even Point Numerically

Using a graphing calculator or software, plot \( P(x) \). Look for the points where the curve crosses the x-axis, indicating break-even points.

Step 4: Analyzing Profit Increase

Inspect the graphed function. As \( x \) increases, observe the behavior of the profit function. Look for a maximum point on the graph, which indicates the largest possible profit.

Step 5: Finding Maximum Profit

Use calculus or a graphing tool to find the critical points by setting the derivative \( P'(x) \) to zero, which gives: \\[ P'(x) = 8 + 0.6x - 0.0039x^2 = 0 \] \Solve this equation to find the x-value that gives the maximum profit.

Key Concepts

Cubic Polynomial

A cubic polynomial is a mathematical expression involving terms up to the third degree, meaning it includes an \(x^3\) term. In our problem, the profit function \(P(x) = 8x + 0.3x^2 - 0.0013x^3 - 372\) is an example of this. The cubic term, \(-0.0013x^3\), causes the graph of the polynomial to have a curve that changes direction. The shape of a cubic polynomial is typically a wave with one or two turning points.
Each term in the polynomial has significance:

• The linear term \(8x\), suggests that profit increases by $8 for each additional blender sold.
• The quadratic term \(0.3x^2\), adjusts this increase, potentially slowing down or speeding up depending on the value of \(x\).
• The cubic term \(-0.0013x^3\), indicates that at some point, too many blenders might actually decrease profit.

This mixture of terms allows the profit function to change dynamically as more blenders are produced and sold.

Break-even Point

The break-even point occurs when total profit is zero, meaning the firm's costs are exactly covered by its revenues. To find the break-even point for our profit function, set \(P(x)\) to zero: \[ 8x + 0.3x^2 - 0.0013x^3 - 372 = 0 \] This equation shows that the profit function is at zero at certain values of \(x\). These \(x\) values are the points where the company neither makes a profit nor a loss.
Finding these points involves solving a cubic equation, which can be complex. Typically, numerical methods or graphing software are used to find solutions.

• Graph the function and look for intersections with the x-axis.
• These intersections represent the production levels where profit equals zero.

Recognizing break-even points helps the company set minimum production goals to avoid losses.

Numerical Solution

Numerical solutions are essential when equations cannot be easily solved analytically. For the break-even equation, exact algebraic solutions are challenging. Instead, numerical methods are employed. These include using technology to approximate where the curve crosses specific points, such as the x-axis.
Graphing calculators or software like Desmos or GeoGebra plot the cubic polynomial and allow us to visually identify these points:

• Observe where the function crosses the x-axis to determine the break-even points.
• Software can provide detailed estimates of these values.

Utilizing numerical solutions helps evaluate complex polynomials without requiring cumbersome calculations.

Maximum Profit Analysis

To determine the maximum profit, analyze the behavior of the profit function as the production level \(x\) changes. Maximum profit occurs at the peak or highest point of the polynomial graph. This involves finding critical points of the function, typically by calculating its derivative \(P'(x)\).
The derivative \(P'(x) = 8 + 0.6x - 0.0039x^2\) provides insights into the rate of change in profit. Setting it to zero helps locate the turning points of the function: \[ 8 + 0.6x - 0.0039x^2 = 0 \] Solving this equation identifies potential points for maximum profit by finding values of \(x\) where the profit function changes from increasing to decreasing.

• Use a graphing tool to verify the maximum profit point on the graph.
• This point is where incremental production starts reducing profit.

Understanding maximum profit assists businesses in optimizing production strategies.