Finding The Vertex Of The Parabola Y=-6x^2+100x-180 For Profit Maximization

In the world of business, understanding the relationship between price, cost, and profit is crucial for success. Mathematical models can often help us to visualize and analyze these relationships, enabling us to make informed decisions. This article delves into a specific scenario involving the sale of soccer balls, where the daily profit ($y$) is modeled by a quadratic equation based on the selling price ($x$) of each soccer ball. We will explore how to determine the vertex of the parabola represented by this equation, which provides valuable insights into the profit-maximizing price point.

This article aims to explore the mathematical concepts behind quadratic equations and their graphical representation as parabolas, and how these concepts can be applied to real-world scenarios. By understanding the vertex of a parabola, we can optimize business decisions, such as determining the ideal selling price for a product to maximize profit. Specifically, we will focus on the equation $y = -6x^2 + 100x - 180$, which models the daily profit ($y$) from soccer ball sales based on the selling price ($x$).

The Quadratic Equation: A Model for Profit

The equation $y = -6x^2 + 100x - 180$ is a quadratic equation, where $y$ represents the daily profit from soccer balls and $x$ represents the selling price of each soccer ball. The coefficients in this equation play a crucial role in determining the shape and position of the parabola. The negative coefficient of the $x^2$ term (-6) indicates that the parabola opens downwards, meaning that there is a maximum point, or vertex. This vertex represents the selling price ($x$) that maximizes daily profit ($y$). The other coefficients, 100 (coefficient of the $x$ term) and -180 (constant term), also influence the position and shape of the parabola, impacting the vertex and overall profit potential.

Understanding the Parabola

When we graph this equation, we obtain a parabola, a U-shaped curve that is symmetrical about a vertical line called the axis of symmetry. The vertex of the parabola is the point where the curve changes direction. In this case, since the parabola opens downwards, the vertex represents the maximum point on the curve, corresponding to the maximum profit. The $x$-coordinate of the vertex tells us the selling price that will yield this maximum profit, and the $y$-coordinate tells us the maximum profit itself. To find the vertex, we can use a formula or complete the square. The formula for the $x$-coordinate of the vertex is given by $-b/(2a)$, where $a$ and $b$ are the coefficients of the quadratic equation. In our equation, $a = -6$ and $b = 100$.

Calculating the Vertex: A Step-by-Step Guide

To find the vertex of the parabola represented by the equation $y = -6x^2 + 100x - 180$, we need to determine the $x$ and $y$ coordinates of the vertex. The $x$-coordinate of the vertex can be found using the formula $x = -b / (2a)$, where $a$ and $b$ are the coefficients of the quadratic equation. In this case, $a = -6$ and $b = 100$. Plugging these values into the formula, we get: $x = -100 / (2 * -6) = -100 / -12 = 8.33$. This means that the selling price that maximizes profit is approximately $8.33. Now that we have the $x$-coordinate, we can find the $y$-coordinate (the maximum profit) by substituting $x = 8.33$ back into the original equation: $y = -6(8.33)^2 + 100(8.33) - 180$. Calculating this gives us: $y = -6(69.3889) + 833 - 180 = -416.3334 + 833 - 180 = 236.6666$. Therefore, the vertex of the parabola is approximately $(8.33, 236.67)$.

Finding the Vertex of $y = -6x^2 + 100x - 180$

The Vertex Formula: A Quick Solution

The vertex of a parabola is a crucial point, representing either the maximum or minimum value of the quadratic function. In the context of our problem, the vertex of the parabola defined by $y = -6x^2 + 100x - 180$ represents the selling price of soccer balls that maximizes daily profit. Since the coefficient of the $x^2$ term is negative (-6), the parabola opens downwards, indicating that the vertex is the highest point on the curve, hence the maximum profit. To find the vertex, we can utilize the vertex formula, a straightforward method derived from the process of completing the square. The formula states that for a quadratic equation in the form $y = ax^2 + bx + c$, the $x$-coordinate of the vertex is given by $x = -b / (2a)$. This formula is a powerful tool that allows us to quickly determine the $x$-value at which the maximum or minimum value of the quadratic function occurs.

In our specific equation, $y = -6x^2 + 100x - 180$, we can identify the coefficients as $a = -6$, $b = 100$, and $c = -180$. Plugging these values into the vertex formula, we get: $x = -100 / (2 * -6) = -100 / -12 = 8.33$. This calculation tells us that the selling price of $8.33 per soccer ball corresponds to the maximum daily profit. To find the actual maximum profit, we need to substitute this $x$-value back into the original equation. This will give us the $y$-coordinate of the vertex, which represents the maximum profit in dollars. Understanding and applying the vertex formula is essential for solving problems involving quadratic functions, especially when optimizing quantities such as profit, cost, or area.

Calculating the Maximum Profit

After determining the optimal selling price ($x = 8.33$) using the vertex formula, the next step is to calculate the maximum profit ($y$) that can be achieved at this price point. To do this, we substitute the value of $x$ back into the original equation: $y = -6x^2 + 100x - 180$. Replacing $x$ with $8.33$, we get: $y = -6(8.33)^2 + 100(8.33) - 180$. Now, we perform the calculations step by step. First, we square $8.33$, which gives us approximately $69.3889$. Then, we multiply this by -6, resulting in approximately $-416.3334$. Next, we multiply 100 by $8.33$, which equals $833$. Finally, we subtract 180 from the sum. So, the equation becomes: $y = -416.3334 + 833 - 180$. Adding and subtracting these values, we get: $y = 236.6666$. Rounding this to the nearest hundredth, we find that the maximum daily profit is approximately $236.67. This value represents the $y$-coordinate of the vertex and signifies the highest profit that can be achieved by selling soccer balls at the price of $8.33 each. Understanding how to calculate the maximum profit is crucial for businesses aiming to optimize their pricing strategies and maximize their earnings. The vertex of the parabola, in this context, provides a clear and concise answer to the question of how to set the price for maximum profitability.

Analyzing the Options: Finding the Correct Vertex

When presented with multiple options for the vertex of a parabola, it's essential to analyze each one carefully to determine the correct answer. We've already calculated the vertex for the equation $y = -6x^2 + 100x - 180$ to be approximately $(8.33, 236.67)$. Now, let's examine the given options and see which one matches our calculation.

The options provided are:

• $(16.67, 1,306.67)$
• $(8.33, 236.67)$
• $(0.03, -177)$
• $(8.33, 0)$
• $(2, 0)$
• $(14.61, 0)$

Evaluating the Options

By comparing these options with our calculated vertex $(8.33, 236.67)$, we can see that only one option matches our result. The option $(16.67, 1,306.67)$ has an $x$-coordinate significantly different from our calculated $x$-coordinate of $8.33$, and the $y$-coordinate is also much larger than our calculated maximum profit of $236.67$. The option $(0.03, -177)$ has an $x$-coordinate close to zero, indicating a very low selling price, and the negative $y$-coordinate suggests a loss rather than a profit. The options $(8.33, 0)$, $(2, 0)$, and $(14.61, 0)$ all have a $y$-coordinate of 0, which means zero profit. This is unlikely to be the vertex, as we know there is a maximum profit to be made. Therefore, these options can be eliminated as well.

The Correct Answer

After careful analysis, it is clear that the correct vertex is $(8.33, 236.67)$. This point represents the selling price of approximately $8.33 per soccer ball, which will yield a maximum daily profit of approximately $236.67. Understanding how to find and interpret the vertex of a parabola is a valuable skill in various applications, including business, engineering, and physics. In the context of this problem, it allows us to optimize the pricing strategy for soccer balls to maximize profit.

Conclusion: Optimizing Profit with Quadratic Equations

In conclusion, understanding the properties of quadratic equations and their graphical representation as parabolas can provide valuable insights for optimizing real-world scenarios. In this article, we explored how to determine the vertex of a parabola represented by the equation $y = -6x^2 + 100x - 180$, which models the daily profit from soccer ball sales based on the selling price. By using the vertex formula, we found that the $x$-coordinate of the vertex is approximately 8.33, representing the optimal selling price of $8.33 per soccer ball. Substituting this value back into the equation, we calculated the $y$-coordinate of the vertex to be approximately 236.67, representing the maximum daily profit of $236.67.

The vertex of a parabola is a critical point, indicating the maximum or minimum value of the quadratic function. In our case, the vertex represented the maximum profit, as the parabola opened downwards due to the negative coefficient of the $x^2$ term. By analyzing the given options for the vertex, we were able to identify the correct answer as $(8.33, 236.67)$, confirming our calculations. This example demonstrates how mathematical concepts, such as quadratic equations and parabolas, can be applied to practical business situations to make informed decisions and optimize outcomes. Understanding the relationship between price, cost, and profit is essential for any business, and mathematical models can provide a powerful tool for analyzing and maximizing these factors. By mastering these concepts, businesses can improve their profitability and achieve greater success in the marketplace.

In summary, the process of finding the vertex involves: identifying the coefficients of the quadratic equation, applying the vertex formula to find the $x$-coordinate, substituting the $x$-coordinate back into the equation to find the $y$-coordinate, and interpreting the results in the context of the problem. This approach can be applied to various optimization problems, making it a valuable skill for anyone involved in decision-making and problem-solving.