Bicycle Company Profit: A Mathematical Exploration

Understanding the Core Components: Revenue and Cost

When we talk about the profit function for a company, we're essentially looking at the difference between how much money they bring in (revenue) and how much money they spend (cost). In the world of business and mathematics, this relationship is fundamental to understanding a company's financial health and decision-making. For our specific case, we're diving into the economics of a bicycle company. The price they receive for each bicycle is determined by a specific equation, and we also know the cost to produce each of those bicycles. To craft the profit function, we need to meticulously combine these two critical elements: the revenue generated and the expenses incurred. Let's break down how we arrive at this crucial function, ensuring we capture all the nuances involved in producing and selling bicycles. The price equation, $b = 100 - 10x^2$, tells us that the price ($b$) a bicycle sells for isn't fixed. It actually decreases as the number of bicycles produced ($x$, in millions) increases. This is an interesting dynamic – perhaps due to market saturation or the need to offer discounts for bulk production. On the other side of the coin, the cost to make each bicycle is a straightforward $60. This means for every single bicycle that rolls off the assembly line, the company spends $60. To get the total cost, we'll need to multiply this per-unit cost by the total number of bicycles produced. By understanding these individual pieces – the revenue side influenced by production volume and the fixed cost per unit – we lay the groundwork for calculating the overall profit.

Deriving the Revenue Function: Price Meets Production

The revenue function is the total income a company generates from selling its products. To calculate this, we multiply the price per unit by the number of units sold. In our bicycle company scenario, the price per bicycle is given by the equation $b = 100 - 10x^2$, where $x$ represents the number of bicycles produced in millions. So, to find the total revenue, we need to multiply this price equation by the number of bicycles produced, $x$. This gives us the revenue function, often denoted as $R(x)$.

Mathematically, this looks like:

$R(x) = \text{price per bicycle} \times \text{number of bicycles produced}$

Substituting the given price equation, we get:

$R(x) = (100 - 10x^2) imes x$

Now, let's distribute the $x$ across the terms in the parenthesis:

$R(x) = 100x - 10x^3$

This equation, $R(x) = 100x - 10x^3$, represents the total revenue the company can expect to earn based on the number of bicycles produced and sold. It's important to note the units here. If $x$ is in millions of bicycles, then $R(x)$ would represent revenue in some monetary unit (e.g., dollars) for millions of bicycles sold. This function captures how the total income changes as production levels fluctuate, taking into account the non-linear relationship between production volume and selling price. The term $-10x^3$ indicates that as production increases significantly, the revenue might not grow linearly and could even start to decrease if the price reduction effect becomes dominant. This is a crucial insight for any company looking to optimize its production.

Calculating the Total Cost Function: The Expense of Production

Moving on to the expenses, the total cost function represents all the money a company spends to produce its goods. In this problem, we're given a simplified scenario where the cost to make each bicycle is a constant $60. This is known as the variable cost per unit, as it directly scales with the number of units produced. To find the total cost, we need to multiply this cost per bicycle by the total number of bicycles produced.

Let's denote the total cost function as $C(x)$. The formula is:

$C(x) = \text{cost per bicycle} \times \text{number of bicycles produced}$

Given that the cost per bicycle is $60 and the number of bicycles produced is $x$ (in millions), the total cost function becomes:

$C(x) = 60 imes x$

So, $C(x) = 60x$.

This equation is quite straightforward. It indicates that the total cost of production is directly proportional to the number of bicycles manufactured. If the company produces 1 million bicycles, the cost is $60 million. If they produce 2 million, the cost is $120 million, and so on. This linear relationship assumes no fixed costs (like factory rent or machinery depreciation) are involved, or that these fixed costs are negligible or already accounted for elsewhere. In a real-world scenario, total cost often includes both fixed costs and variable costs. However, based on the information provided in the problem, our total cost function is simply $C(x) = 60x$. This function is vital because it quantifies the financial outlay required to bring the bicycles to market, and it will be subtracted from the revenue to determine the profit.

Unveiling the Profit Function: Revenue Minus Cost

Now that we have successfully derived both the revenue function ($R(x)$) and the total cost function ($C(x)$), we can finally construct the profit function. The profit function, often denoted as $P(x)$, is the cornerstone of our analysis. It tells us the company's net earnings after accounting for all expenses. The fundamental relationship is:

$P(x) = R(x) - C(x)$

We have already established that:

$R(x) = 100x - 10x^3$

And:

$C(x) = 60x$

Now, we substitute these expressions into the profit function formula:

$P(x) = (100x - 10x^3) - (60x)$

To simplify this expression, we remove the parentheses and combine like terms. We need to be careful with the subtraction of the cost function:

$P(x) = 100x - 10x^3 - 60x$

Combining the terms involving $x$ (100x and -60x):

$P(x) = (100 - 60)x - 10x^3$

$P(x) = 40x - 10x^3$

This is the profit function for the bicycle company. It represents the net profit the company makes based on the number of bicycles produced, $x$ (in millions). This function is crucial because it allows the company to analyze its profitability at different production levels. For instance, if they produce $x=1$ million bicycles, the profit would be $P(1) = 40(1) - 10(1)^3 = 40 - 10 = 30$ (in millions of dollars, assuming). If they produce $x=2$ million bicycles, $P(2) = 40(2) - 10(2)^3 = 80 - 10(8) = 80 - 80 = 0$. This indicates that at 2 million units, the company breaks even. It's important to note that the problem statement provided options for the profit function, and our derived function $P(x) = 40x - 10x^3$ does not directly match any of the given options A or B if interpreted strictly. However, if we re-examine the options, there might be a misunderstanding or a typo in the provided options or the question's premise. Let's assume for a moment there was a typo and the price equation was intended differently, or the options are derived from a slightly different cost structure. Let's re-evaluate the original question and options carefully.

Analyzing the Provided Options and Refining the Solution

Let's revisit the original problem statement and the provided options for the profit function: The price received for a bicycle is given by the equation $b = 100 - 10x^2$, where $x$ is the number of bicycles produced, in millions. It costs the company $60 to make each bicycle. What is the profit function for the company?

Option A: $P = -10x^2 - 60x + 100$ Option B: (This seems to be a placeholder for a second option, but it's not provided).

Our derived profit function is $P(x) = 40x - 10x^3$. This function is derived directly from the stated revenue and cost components. Let's meticulously check our steps.

Revenue: $R(x) = ( ext{price per bicycle}) imes ( ext{number of bicycles})$ $R(x) = (100 - 10x^2) imes x$ $R(x) = 100x - 10x^3$

Cost: $C(x) = ( ext{cost per bicycle}) imes ( ext{number of bicycles})$ $C(x) = 60 imes x$ $C(x) = 60x$

Profit: $P(x) = R(x) - C(x)$ $P(x) = (100x - 10x^3) - (60x)$ $P(x) = 100x - 10x^3 - 60x$ $P(x) = 40x - 10x^3$

It appears there might be a discrepancy between our derived function and the options provided. Let's consider if the question or the options intended a different interpretation.

• Could the price equation be interpreted differently? The equation $b=100-10x^2$ clearly defines the price per bicycle as a function of the quantity produced. This is a common economic model where price is dependent on supply.
• Could the cost be interpreted differently? The statement "It costs the company $60 to make each bicycle" is generally interpreted as a constant variable cost per unit.

If we strictly follow the given information and standard economic definitions, our profit function is indeed $P(x) = 40x - 10x^3$. None of the provided options A or B seem to align with this result.

Let's hypothesize potential misinterpretations or typos that could lead to one of the options.

Hypothetical Scenario 1: Typo in the price equation. If the price equation was $b = 100 - 10x$ (linear relationship instead of quadratic in $x^2$), then: $R(x) = (100 - 10x)x = 100x - 10x^2$ $P(x) = R(x) - C(x) = (100x - 10x^2) - 60x = 40x - 10x^2$. This still doesn't match option A, which is $P = -10x^2 - 60x + 100$.

Hypothetical Scenario 2: Typo in the cost structure, or the option includes fixed costs or revenue differently. Let's examine Option A: $P = -10x^2 - 60x + 100$. This function has an $x^2$ term, a linear $x$ term, and a constant term. Our derived $P(x)$ has a cubic $x^3$ term and a linear $x$ term.

If the revenue was $R(x) = 100 - 10x^2$ (which would mean revenue is constant or decreases with production, which is unusual for a revenue function itself unless it represents profit directly in some context), and the cost was $C(x) = 60x$, then $P(x) = (100 - 10x^2) - 60x = -10x^2 - 60x + 100$. This exactly matches Option A. However, this requires interpreting the initial equation $b=100-10x^2$ not as the price per bicycle, but as the total revenue function $R(x)$ itself, and that the cost is $C(x)=60x$. This is a significant reinterpretation of the problem statement.

Given the phrasing "The price received for a bicycle is given by the equation $b=100-10x^2$", it is most natural to interpret $b$ as the price per unit. Therefore, our initial derivation $P(x) = 40x - 10x^3$ is the mathematically sound answer based on the provided text.

However, if forced to choose from the given options and assuming there might be an intended interpretation leading to one of them, the only way to arrive at Option A ($P = -10x^2 - 60x + 100$) is by assuming that the equation $b=100-10x^2$ represents the total revenue $R(x)$ and that the cost function $C(x)$ is $60x$. This interpretation contradicts the explicit statement that $b$ is the price received for a bicycle.

It's highly probable there's an error in the question's provided options or its wording. Assuming the most standard interpretation of the problem as written, the correct profit function is $P(x) = 40x - 10x^3$. If we must select from the options and assume a different, less direct interpretation, Option A can be obtained.

Let's assume, for the sake of providing an answer aligned with one of the choices, that the question intended for $R(x) = 100 - 10x^2$ and $C(x) = 60x$. This is a highly unconventional interpretation of "price received for a bicycle is given by the equation $b=100-10x^2$" where $x$ is the number of bicycles. It would imply that the total revenue is described by this quadratic equation, which makes little sense economically as revenue should increase with quantity in most cases, or at least have a variable tied to quantity in its primary form. Nevertheless, if we proceed with this assumption:

$R(x) = 100 - 10x^2$ $C(x) = 60x$ $P(x) = R(x) - C(x) = (100 - 10x^2) - (60x)$ $P(x) = 100 - 10x^2 - 60x$ Rearranging terms to the standard polynomial form (highest degree first): $P(x) = -10x^2 - 60x + 100$

This result exactly matches Option A.

Therefore, if we are compelled to choose from the given options, Option A is the only one that can be reached, albeit through a questionable reinterpretation of the problem statement where $b=100-10x^2$ is treated as the total revenue function $R(x)$ and the cost function is $C(x)=60x$.

Conclusion: Navigating Economic Models and Mathematical Functions

Understanding the profit function is paramount for any business aiming for sustainability and growth. It serves as a diagnostic tool, revealing the financial impact of production and sales strategies. In this exploration of a bicycle company, we meticulously derived the profit function by first defining the revenue and cost components. The revenue, calculated as the price per bicycle multiplied by the number of units sold, was found to be $R(x) = 100x - 10x^3$, based on the given price equation $b = 100 - 10x^2$. The cost function, determined by the fixed cost per bicycle, was $C(x) = 60x$. Subtracting the cost from the revenue yielded our primary derived profit function: $P(x) = 40x - 10x^3$.

However, when comparing this derived function to the provided options, a discrepancy was noted. Option A, $P = -10x^2 - 60x + 100$, can only be obtained if we assume a significant reinterpretation of the problem statement, specifically treating the given price equation $b = 100 - 10x^2$ as the total revenue function $R(x)$ directly, and maintaining the cost function $C(x) = 60x$. This interpretation is less aligned with the explicit wording that $b$ represents the price received for a bicycle.

This scenario highlights the importance of precise language in mathematical and economic problems. While our initial, most direct derivation leads to $P(x) = 40x - 10x^3$, the existence of specific options often implies an intended interpretation, even if it requires bending the literal meaning. In educational contexts, it's crucial to understand both the correct derivation and how to navigate potential ambiguities or errors in question design.

For further reading on economic principles and the application of calculus in business, you can explore resources on economic modeling and calculus applications in business.