Mario's Lawn Mowing Profit Calculation And Analysis

In this article, we delve into the mathematical intricacies of Mario's lawn mowing business. Mario, an enterprising individual, earns money by mowing his neighbors' lawns. To understand the financial dynamics of his venture, we will analyze his revenue, costs, and ultimately, his profit. The scenario presents a classic example of applying mathematical functions to model real-world business situations. We will explore how Mario's profit is determined by the number of lawns he mows, taking into account his earnings and expenses. By examining the functions representing his revenue and costs, we can derive a profit function that will allow us to calculate his earnings for any given number of lawns mowed.

The core of this analysis lies in understanding the relationship between revenue, cost, and profit. Revenue is the income Mario generates from mowing lawns, while cost represents his expenses, such as gas and mower rental. Profit, the ultimate goal of any business, is the difference between revenue and cost. By expressing these elements as mathematical functions, we can gain a clear picture of Mario's financial performance. This exploration not only provides a practical application of mathematical concepts but also highlights the importance of financial literacy in everyday life. We will dissect the provided functions, r(x) for revenue and c(x) for cost, to derive p(x), the profit function. This journey will involve basic algebraic operations and a clear understanding of function notation. Through this analysis, we aim to provide a comprehensive understanding of how Mario can maximize his profit and make informed decisions about his lawn mowing business.

To begin our analysis of Mario's lawn mowing business, we must first define the functions that govern his revenue, cost, and profit. Revenue, denoted as r(x), represents the total income Mario earns from mowing x lawns. The problem states that r(x) = 25x. This means that for each lawn Mario mows, he earns $25. The revenue function is a linear function, indicating a direct proportionality between the number of lawns mowed and the income generated. This simple yet crucial function sets the foundation for understanding Mario's earning potential.

Next, we consider Mario's costs, represented by the function c(x). These costs encompass the expenses Mario incurs in operating his business, specifically the cost of gas and mower rental. The given function is c(x) = 6x + 15. This function is also linear, but it includes both a variable cost component (6x) and a fixed cost component (15). The variable cost, 6x, implies that Mario spends $6 on gas for each lawn he mows. The fixed cost, $15, likely represents the rental fee for the mower, which remains constant regardless of the number of lawns mowed. Understanding the cost function is essential for determining the profitability of Mario's business, as it directly impacts his net earnings.

Finally, we arrive at the profit function, p(x), which is the central focus of our analysis. Profit is the difference between revenue and cost, representing the actual earnings Mario retains after covering his expenses. Mathematically, this is expressed as p(x) = (r - c)(x), which means p(x) = r(x) - c(x). To determine the explicit form of p(x), we will substitute the given expressions for r(x) and c(x) and simplify the resulting expression. This will provide us with a clear formula for calculating Mario's profit based on the number of lawns he mows. The profit function is the key to understanding the financial viability of Mario's business and will allow us to answer the question posed in the problem.

The central question we aim to answer is: What is p(x), Mario's profit function? As established earlier, the profit function, p(x), is the difference between the revenue function, r(x), and the cost function, c(x). Mathematically, this is expressed as:

p(x) = r(x) - c(x)

We are given that Mario's revenue function is r(x) = 25x, which means he earns $25 for every lawn he mows. His cost function, c(x), is given as c(x) = 6x + 15, representing his expenses for gas and mower rental. To find the profit function, we substitute these expressions into the equation:

p(x) = (25x) - (6x + 15)

Now, we need to simplify this expression. The first step is to distribute the negative sign across the terms within the parentheses:

p(x) = 25x - 6x - 15

Next, we combine the like terms, which in this case are the terms involving x:

p(x) = (25x - 6x) - 15

This simplifies to:

p(x) = 19x - 15

Therefore, Mario's profit function, p(x), is 19x - 15. This function tells us that for every lawn Mario mows, he makes a profit of $19 after accounting for his expenses, but he has an initial cost of $15 (presumably for the mower rental). This profit function is crucial for Mario to understand his earnings potential and to make informed decisions about his business. For instance, he can use this function to determine how many lawns he needs to mow to reach a specific profit target. Understanding the profit function is not just about calculating numbers; it's about empowering Mario to manage his business effectively and achieve his financial goals.

Now that we have derived Mario's profit function, p(x) = 19x - 15, we can delve into analyzing its implications for his lawn mowing business. This function provides valuable insights into Mario's profitability and can help him make strategic decisions.

The profit function is a linear equation, which means it has a constant rate of change. In this case, the coefficient of x, which is 19, represents the profit Mario earns for each additional lawn he mows. This is his marginal profit. The constant term, -15, represents the fixed cost, which is the initial expense Mario incurs regardless of the number of lawns he mows. This is likely the cost of renting the mower.

To understand the significance of this function, let's consider a few scenarios. If Mario mows zero lawns (x = 0), his profit would be:

p(0) = 19(0) - 15 = -15

This indicates that Mario would have a loss of $15 if he doesn't mow any lawns, which is the cost of the mower rental. This highlights the importance of mowing enough lawns to cover his fixed costs.

To determine how many lawns Mario needs to mow to break even (i.e., make zero profit), we set p(x) = 0 and solve for x:

0 = 19x - 15

19x = 15

x = 15 / 19 ≈ 0.79

Since Mario cannot mow a fraction of a lawn, he needs to mow at least one lawn to start making a profit. In reality, he'd need to mow more than one lawn to truly cover the $15 expense and start seeing a positive return.

If Mario mows 10 lawns, his profit would be:

p(10) = 19(10) - 15 = 190 - 15 = 175

This means Mario would earn a profit of $175 if he mows 10 lawns. This illustrates the potential for profit as Mario mows more lawns. The profit function provides a clear and concise way for Mario to estimate his earnings based on his workload.

Furthermore, the profit function can help Mario assess the impact of changes in his costs or revenue. For example, if the cost of gas increases, this would affect the coefficient of x in the cost function, which in turn would alter the profit function. Similarly, if Mario decides to charge more per lawn, this would change the coefficient of x in the revenue function, again impacting his profit function. Understanding these relationships allows Mario to adapt his business strategy to maximize his profitability. In essence, the profit function is a powerful tool that empowers Mario to make informed decisions and manage his lawn mowing business effectively.

In conclusion, the analysis of Mario's lawn mowing business provides a compelling illustration of the power of mathematical modeling in understanding and managing real-world business scenarios. By defining and analyzing the revenue, cost, and profit functions, we have gained valuable insights into the financial dynamics of Mario's venture.

The revenue function, r(x) = 25x, highlights the direct relationship between the number of lawns mowed and the income generated. The cost function, c(x) = 6x + 15, reveals the expenses involved, including both variable costs (gas) and fixed costs (mower rental). The profit function, p(x) = 19x - 15, which we derived by subtracting the cost function from the revenue function, provides a clear picture of Mario's earnings after accounting for his expenses.

The profit function is not just a mathematical formula; it's a powerful tool that Mario can use to make informed decisions. By understanding the slope and intercept of this linear function, Mario can quickly estimate his profit for any given number of lawns mowed. He can also determine the number of lawns he needs to mow to break even or to achieve a specific profit target. Furthermore, the profit function allows Mario to assess the impact of changes in his costs or revenue, such as an increase in gas prices or a decision to charge more per lawn.

This analysis demonstrates the practical application of mathematical concepts in everyday business situations. By using functions to model revenue, cost, and profit, we can gain a deeper understanding of the underlying financial principles. This knowledge empowers individuals like Mario to manage their businesses effectively and to make strategic decisions that maximize their profitability.

More broadly, this example underscores the importance of financial literacy and mathematical skills in various aspects of life. Whether it's managing personal finances, making investment decisions, or running a small business, a solid understanding of mathematical concepts can be invaluable. By applying mathematical principles, we can make more informed choices and achieve our financial goals. Mario's lawn mowing business serves as a simple yet insightful case study, showcasing the transformative power of mathematical modeling in the world of business and beyond.

Answer: The profit function p(x) is:

p(x) = 19x - 15

So the correct answer would be an option with this expression.