Problem 81 - Precalculus Mathematics for Calculus

Question

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) The firm breaks even at producing approximately 60 blenders. (b) The largest possible profit is about $448 when producing around 230 blenders.

Step 1: Understand the Profit Function

The profit function provided is \( P(x) = 8x + 0.3x^2 - 0.0013x^3 - 372 \). This function is a cubic polynomial, where \( x \) represents the number of blenders produced and sold.

Step 2: Finding Break-even Points

To find the break-even point where the profit is zero, set \( P(x) = 0 \): \[ 8x + 0.3x^2 - 0.0013x^3 - 372 = 0 \]This equation can be solved numerically or graphically to find the values of \( x \) where the profit is zero. We'll use a calculator or software to find \( x \) values approximately.

Step 3: Graph the Profit Function

Graph the function \( P(x) = 8x + 0.3x^2 - 0.0013x^3 - 372 \) over a reasonable range of \( x \), such as from \( x = 0 \) to \( x = 1000 \). Use graphing software or a calculator to visualize the behavior of the function.

Step 4: Determine Break-even Points from Graph

From the graph, identify the \( x \)-coordinate (blenders manufactured) where the curve crosses the \( x \)-axis, this represents the break-even point(s). We can interpolate from the graph when \( x \approx 60 \) as a break-even point.

Step 5: Investigate Maximum Profit

Since the function is cubic with a negative leading coefficient, the curve will eventually decrease after achieving a maximum point. Use a graph to estimate the \( x \) value at which the maximum profit occurs (where the slope of the tangent is zero). The maximum is found visually or using a calculator to be around \( x \approx 230 \), with maximum profit approximately \( \$448 \).

Step 6: Analyze Profit Behavior

The profit function decreases after a certain point indicating it will not increase indefinitely. Therefore, the maximum profit is the largest value on the graph where the curve stops rising.

Key Concepts

Cubic Polynomials and Profit Functions

A cubic polynomial is a type of polynomial function with a degree of three. This means its highest power of the variable is cubed. The general form of a cubic polynomial is \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \). In the context of profit functions like the one given, \( P(x) = 8x + 0.3x^2 - 0.0013x^3 - 372 \), the cubic polynomial models how profit changes with the number of items (in this case, blenders) produced and sold.

Cubic polynomials can have various characteristics based on the coefficients and constants:

• A negative leading coefficient, like \(-0.0013\) in this function, indicates that while the profit may increase initially, it will eventually decrease as \(x\) grows.
• The shape of the graph primarily depends on the sign of the leading term and shifts according to the other terms.

Understanding these characteristics helps in predicting the behavior of a profit function over different production levels.

Identifying Break-even Points

The break-even point is a critical concept in business and finance. It represents the level of production and sales at which total revenues equal total costs, resulting in zero profit. For the given profit function, the task is to find the values of \(x\) (the number of blenders) where \(P(x) = 0\).

Setting \(8x + 0.3x^2 - 0.0013x^3 - 372 = 0\) allows us to find the break-even points. This can be done by graphically plotting the function and observing where it crosses the horizontal axis:

• The graph intersects the \(x\)-axis at the break-even points.
• For this problem, plotting the function reveals that the break-even occurs approximately at \(x \approx 60\).

Therefore, calculating or plotting these points is essential for businesses to understand how many items need to be sold to avoid losses.

Graphing Functions to Analyze Profit Behavior

Graphing functions is a powerful visual method to understand and analyze various behaviors and trends. The formula \(P(x) = 8x + 0.3x^2 - 0.0013x^3 - 372\) is best understood when visually represented on a graph. Such a graph displays how the profit changes with the number of blenders produced. Here's how to effectively use graphing for analysis:

• Select a suitable range, such as from \(x = 0\) to \(x = 1000\), to include potential values for production levels.
• The graph helps identify the maximum profit point where the curve reaches its highest level before declining due to the cubic nature of the polynomial.
• The point where the slope of the curve is zero indicates the maximum profit, observed at around \(x \approx 230\) with an estimated profit of \$448.

By examining the graph, businesses can effectively determine the optimal level of production to maximize profits and identify potential losses beyond a specific production level.