Problem 25 - Algebra Form and Function

Question

The profit from selling \(q\) items of a certain product is \(P(q)=36 q-0.0001 q^{3}\) dollars. Find the values of \(q\) such that \(P(q)=0\). Which of these values make sense in the context of the problem? Interpret the values that make sense.

Step-by-Step Solution

Answer

Answer: The realistic values of q for which the profit function P(q) = 36q - 0.0001q^3 equals zero are q = 0 and q = 600. These values represent the points where the profit is zero. For q = 0, no items are sold, and the profit is naturally zero. For q = 600, selling 600 items of the product results in zero profit, which could mean that it's the break-even point, where the company's revenue equals its expenses.

Step 1: Analyze the Profit Function

The given profit function is a polynomial of degree 3, given by P(q) = 36q - 0.0001q^3. Our goal is to find the values of q for which P(q) = 0.

Step 2: Set Up the Equation

To find the values of q for which P(q) = 0, we set the profit function equal to 0: 36q - 0.0001q^3 = 0

Step 3: Solve for Possible Values of q

Now we will solve the equation for q. First, we factor out a q: q(36 - 0.0001q^2) = 0 The equation is equal to 0 when either of these factors is equal to 0: 1. q = 0 2. 36 - 0.0001q^2 = 0 We already have the first value q = 0. Next, we'll solve for q in the second equation: 0.0001q^2 = 36 Now, we'll divide both sides by 0.0001: q^2 = 36 / 0.0001 q^2 = 360000 To find the possible values of q, we'll take the square root of both sides: q = ±sqrt(360000) q ≈ ±600 So, we have three possible values for q: 0, 600, and -600.

Step 4: Analyze the Realistic Values of q

We are looking for values of q that make sense in the context of the problem. The fact that q represents the quantity of items sold implies that q must be a non-negative value (i.e., q ≥ 0). This means that the value q = -600 does not make sense in this scenario.

Step 5: Interpret the Realistic Values

The two remaining values of q that make sense are q = 0 and q = 600. These values represent the points where the profit function equals zero. For q = 0, no items are sold, so it is natural for the profit to be zero. For q = 600, selling 600 items of the product results in zero profit. This could mean that it's the break-even point, where the company's revenue equals its expenses. The realistic values for q in terms of the problem's context are q = 0 and q = 600.

Key Concepts

Profit Function

In the business world, understanding profit is crucial for any company. A profit function is a mathematical representation that helps determine the profitability of selling a certain number of products. For this exercise, the profit function is given by the equation \(P(q) = 36q - 0.0001q^3\). This equation falls under the category of cubic functions due to the presence of the term \(-0.0001q^3\).

A profit function like this one can tell us several important things:

• The term \(36q\) represents the revenue, which is the money earned by selling \(q\) units of a product.
• The term \(-0.0001q^3\) represents costs or decreasing returns, which could be related to factors like production costs increasing at higher quantities or limitations in scaling production.

By setting the profit function equal to zero, businesses aim to find the quantity of products where their operations neither gain nor lose money. This is critical for making informed decisions on production and sales strategies.

Polynomial Equation

The equation we are dealing with is known as a polynomial equation. Specifically, it's a cubic polynomial because the highest power of \(q\) is three. The general form of a cubic polynomial is \(ax^3 + bx^2 + cx + d = 0\). In this exercise, the polynomial is \(36q - 0.0001q^3 = 0\).

Polynomial equations are solved by finding the values of the variable(s) that make the equation true. Here, the process involved:

• Factoring out the common factor \(q\) gives us the equation \(q(36 - 0.0001q^2) = 0\).
• This leads to two simpler equations: \(q = 0\) and \(36 - 0.0001q^2 = 0\).
• The second equation is solved further to find that \(q^2 = 360000\), leading to \(q = ±600\).

This process of finding the roots, or solutions, is fundamental in algebra and crucial for tasks like optimizing profit where we're interested in specific outcomes.

Break-even Point

The concept of a break-even point is important in financial assessments. It refers to the situation where total costs and total revenue are equal, resulting in zero profit. Here, the break-even point occurs at values of \(q\) where the profit function \(P(q)\) equals zero.

Analyzing the exercise, two points were found where the profit is zero: \(q = 0\) and \(q = 600\). Understanding these:

• At \(q = 0\), no items are sold, hence no profit is made, which is a trivial break-even point.
• At \(q = 600\), the company reaches a substantial break-even point. It suggests they must sell 600 items of their product to cover all costs, leading to zero profit.

This is crucial for businesses as it establishes a target volume of sales that they need to achieve to cover costs. Knowing the break-even point helps companies make decisions about pricing, budgets, and production levels.