Finding Zeros Of Y=-6x^2+100x-180 And Profit Maximization
Introduction
In the dynamic world of sports equipment retail, understanding the relationship between pricing and profit is crucial for success. For a business selling soccer balls, the daily profit (y) often depends on the selling price (x) of each ball. This relationship can sometimes be modeled by a quadratic equation, allowing us to analyze and optimize pricing strategies. In this article, we will delve into the intricacies of the quadratic equation y = -6x² + 100x - 180, which represents the daily profit from soccer balls based on their selling price. Our primary goal is to determine the zeros of this function, which are the selling prices at which the daily profit is zero. These zeros are critical points, indicating the prices at which the business neither makes a profit nor incurs a loss. By finding these values, we can gain valuable insights into the price range within which the business can operate profitably. We will explore the mathematical methods used to find these zeros, rounding our answers to the nearest hundredth for practical application. Understanding these concepts is not only beneficial for businesses in the sports retail sector but also provides a broader understanding of how quadratic equations can model real-world scenarios and aid in decision-making processes. The journey through this equation will highlight the importance of mathematical analysis in business strategy and demonstrate how seemingly abstract concepts can have tangible implications in the marketplace. Furthermore, we will discuss the significance of these zeros in the context of profit maximization, providing a comprehensive understanding of how pricing strategies can impact a business's bottom line. This exploration will empower readers with the knowledge to interpret and apply quadratic equations in various business contexts, ultimately fostering a more data-driven approach to decision-making.
Understanding the Quadratic Equation
The quadratic equation y = -6x² + 100x - 180 is a powerful mathematical tool for modeling various real-world phenomena, including the relationship between the selling price of a product and the resulting profit. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we are trying to solve for. In our specific equation, a = -6, b = 100, and c = -180. The negative coefficient of the x² term (-6) indicates that the parabola opens downwards, meaning that the profit function has a maximum point. This is a crucial piece of information as it tells us that there is an optimal selling price that will maximize the daily profit. The coefficients b and c play significant roles in determining the shape and position of the parabola. The b term (100) influences the axis of symmetry, which is the vertical line that passes through the vertex (the maximum or minimum point) of the parabola. The c term (-180) represents the y-intercept, which is the point where the parabola intersects the y-axis. In the context of our problem, the y-intercept represents the profit when the selling price (x) is zero. Understanding these coefficients and their roles is essential for interpreting the behavior of the quadratic function and extracting meaningful insights. The zeros of the quadratic equation, also known as the roots or solutions, are the values of x that make y equal to zero. These points are where the parabola intersects the x-axis and represent the selling prices at which the daily profit is zero. Finding these zeros is crucial for determining the price range within which the business can operate profitably. In the following sections, we will explore the methods for finding these zeros and discuss their implications for pricing strategy and profit maximization. By thoroughly understanding the components of the quadratic equation, we can effectively analyze the relationship between selling price and profit and make informed decisions to optimize business outcomes.
Methods for Finding Zeros
To find the zeros of the quadratic equation y = -6x² + 100x - 180, we need to solve for x when y = 0. There are several methods for solving quadratic equations, each with its own advantages and applications. The two most common methods are factoring and using the quadratic formula. Factoring involves rewriting the quadratic equation as a product of two binomials. If we can factor the equation, we can then set each factor equal to zero and solve for x. However, factoring is not always straightforward, especially when the coefficients are large or the roots are not rational numbers. In such cases, the quadratic formula provides a more reliable approach. The quadratic formula is a general solution for quadratic equations of the form ax² + bx + c = 0 and is given by: x = (-b ± √(b² - 4ac)) / (2a). This formula can be used to find the zeros of any quadratic equation, regardless of whether it can be factored. In our case, a = -6, b = 100, and c = -180. Plugging these values into the quadratic formula, we get: x = (-100 ± √(100² - 4*(-6)(-180))) / (2(-6)). Simplifying this expression, we have: x = (-100 ± √(10000 - 4320)) / (-12). Further simplification yields: x = (-100 ± √5680) / (-12). Now, we can calculate the square root of 5680, which is approximately 75.3658. Therefore, we have two possible solutions for x: x₁ = (-100 + 75.3658) / (-12) and x₂ = (-100 - 75.3658) / (-12). Calculating these values, we get: x₁ ≈ 2.05 and x₂ ≈ 14.61. These two values represent the zeros of the quadratic equation and correspond to the selling prices at which the daily profit is zero. In the next section, we will discuss the significance of these zeros in the context of profit maximization and pricing strategy. By understanding the methods for finding zeros and their practical implications, businesses can make informed decisions to optimize their pricing and maximize their profitability.
Calculating the Zeros
In the previous section, we introduced the quadratic formula as a reliable method for finding the zeros of a quadratic equation. Now, let's apply this formula to our specific equation, y = -6x² + 100x - 180, and calculate the zeros to the nearest hundredth. As we established, the quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a), where a = -6, b = 100, and c = -180. Plugging these values into the formula, we get: x = (-100 ± √(100² - 4*(-6)(-180))) / (2(-6)). Simplifying the expression under the square root: 100² - 4*(-6)*(-180) = 10000 - 4320 = 5680. So, the formula becomes: x = (-100 ± √5680) / (-12). Now, we need to calculate the square root of 5680, which is approximately 75.3658. Therefore, we have two possible solutions for x: x₁ = (-100 + 75.3658) / (-12) and x₂ = (-100 - 75.3658) / (-12). Let's calculate x₁: x₁ = (-100 + 75.3658) / (-12) = -24.6342 / -12 ≈ 2.05. Rounding to the nearest hundredth, we get x₁ ≈ 2.05. Now, let's calculate x₂: x₂ = (-100 - 75.3658) / (-12) = -175.3658 / -12 ≈ 14.61. Rounding to the nearest hundredth, we get x₂ ≈ 14.61. These two values, 2.05 and 14.61, represent the zeros of the quadratic equation. In the context of our problem, these zeros represent the selling prices at which the daily profit from soccer balls is zero. This means that if the soccer balls are sold at a price of approximately $2.05 or $14.61, the business will neither make a profit nor incur a loss. These points are crucial for understanding the break-even points of the business. In the next section, we will discuss the implications of these zeros for profit maximization and pricing strategy. By accurately calculating the zeros of the quadratic equation, we gain valuable insights into the financial dynamics of the business and can make informed decisions to optimize profitability.
Interpreting the Zeros
Having calculated the zeros of the quadratic equation y = -6x² + 100x - 180 to be approximately 2.05 and 14.61, it is crucial to interpret these values in the context of the problem. The zeros, also known as the roots or solutions of the equation, represent the selling prices (x) at which the daily profit (y) from soccer balls is zero. In other words, these are the break-even points for the business. When the selling price is $2.05 or $14.61, the business neither makes a profit nor incurs a loss. Understanding these break-even points is essential for making informed decisions about pricing strategy. Selling prices below $2.05 or above $14.61 will result in a loss, as the daily profit (y) will be negative. Conversely, selling prices between $2.05 and $14.61 will result in a profit, as the daily profit (y) will be positive. The quadratic equation represents a parabola that opens downwards, due to the negative coefficient of the x² term (-6). This means that there is a maximum point on the parabola, which corresponds to the selling price that maximizes the daily profit. The x-coordinate of this maximum point is the vertex of the parabola, which can be found using the formula x = -b / (2a). In our case, a = -6 and b = 100, so the x-coordinate of the vertex is x = -100 / (2*(-6)) ≈ 8.33. This means that the selling price that maximizes the daily profit is approximately $8.33. The corresponding maximum profit can be found by plugging this value back into the quadratic equation: y = -6*(8.33)² + 100*(8.33) - 180 ≈ 236.33. Therefore, the maximum daily profit is approximately $236.33 when the selling price is $8.33. By interpreting the zeros in conjunction with the vertex of the parabola, we gain a comprehensive understanding of the relationship between selling price and profit. This information is invaluable for setting optimal pricing strategies, forecasting profits, and making strategic business decisions. In the next section, we will discuss how to use this information to maximize profit and ensure the long-term success of the business.
Implications for Profit Maximization
The zeros of the quadratic equation, along with the vertex, provide critical information for profit maximization. As we've established, the zeros (approximately $2.05 and $14.61) represent the break-even points, while the vertex (approximately $8.33) represents the selling price that maximizes daily profit. Understanding these points allows a business to strategically set prices to achieve its financial goals. To maximize profit, the business should aim to sell soccer balls at a price close to the vertex, which is $8.33. This price will yield the highest daily profit, approximately $236.33. However, it's essential to consider other factors that may influence pricing decisions, such as market demand, competition, and production costs. Setting the price too high, even if it's close to the vertex, may reduce the number of units sold, potentially decreasing overall profit. Conversely, setting the price too low, even if it's above the lower break-even point, may increase sales volume but decrease the profit margin per unit. A comprehensive pricing strategy should also consider the elasticity of demand, which measures how responsive the quantity demanded is to a change in price. If the demand for soccer balls is highly elastic, a small increase in price may lead to a significant decrease in demand. In this case, the business may need to set the price slightly lower than the vertex to maintain sales volume. On the other hand, if the demand is inelastic, the business may have more flexibility in setting prices closer to the vertex without significantly impacting sales. In addition to pricing, businesses can also focus on reducing production costs to increase profitability. Lowering costs shifts the profit curve upwards, increasing the maximum profit and potentially widening the range of profitable selling prices. By analyzing the quadratic equation in conjunction with market dynamics and cost considerations, businesses can develop a robust pricing strategy that maximizes profit and ensures long-term financial sustainability. The insights gained from understanding the zeros and vertex of the quadratic equation provide a solid foundation for making informed decisions and achieving business success. Ultimately, the goal is to strike a balance between price, volume, and cost to optimize profitability and maintain a competitive edge in the market.
Conclusion
In conclusion, the quadratic equation y = -6x² + 100x - 180 provides a valuable model for understanding the relationship between the selling price of soccer balls and daily profit. By finding the zeros of this equation, we identified the break-even points, which are the selling prices at which the business neither makes a profit nor incurs a loss. These zeros, approximately $2.05 and $14.61, define the price range within which the business can operate profitably. Furthermore, by analyzing the vertex of the parabola, we determined the selling price that maximizes daily profit, which is approximately $8.33. At this price, the business can achieve a maximum daily profit of approximately $236.33. The process of finding and interpreting the zeros of the quadratic equation highlights the importance of mathematical analysis in business decision-making. By understanding the underlying mathematical relationships, businesses can make informed decisions about pricing strategy, inventory management, and overall financial planning. The quadratic equation is just one example of how mathematical models can be used to analyze and optimize business operations. Other mathematical tools, such as linear programming, regression analysis, and statistical modeling, can also provide valuable insights into various aspects of business. In the dynamic and competitive world of sports equipment retail, a data-driven approach is essential for success. By leveraging mathematical models and analytical techniques, businesses can gain a competitive edge, improve profitability, and ensure long-term sustainability. The insights gained from analyzing the quadratic equation provide a solid foundation for developing a comprehensive pricing strategy that considers market demand, competition, and cost factors. Ultimately, the ability to interpret and apply mathematical concepts is a valuable asset for any business professional, empowering them to make informed decisions and drive business success. This exploration of the quadratic equation and its implications for soccer ball profitability underscores the power of mathematical thinking in the business world and the importance of embracing a data-driven approach to decision-making.
