Finding Zeroes And Maximizing Profit For The Function Y=-6x^2+100x-180

In the dynamic world of business, understanding the interplay between pricing, costs, and profits is paramount. For entrepreneurs venturing into the sports equipment market, particularly the soccer ball sector, a keen grasp of these factors can be the difference between success and failure. This article delves into the intricate relationship between the selling price of soccer balls and the resulting daily profit, employing the mathematical model $y = -6x^2 + 100x - 180$ as our guide. Here, $x$ represents the selling price of each soccer ball, and $y$ denotes the corresponding daily profit. Our primary objective is to unravel the zeroes of this function, rounding our findings to the nearest hundredth. This exploration will not only illuminate the prices at which the business breaks even but also provide invaluable insights into profit maximization strategies.

Deciphering the Profit Equation: $y = -6x^2 + 100x - 180$

The equation $y = -6x^2 + 100x - 180$ is more than just a mathematical expression; it's a window into the financial health of our hypothetical soccer ball business. This quadratic equation models the daily profit ($y$) as a function of the selling price of each soccer ball ($x$). The negative coefficient of the $x^2$ term (-6) indicates that the profit curve is a parabola that opens downwards. This implies that there's a maximum profit point, beyond which increasing the selling price will actually decrease the daily profit. The other terms, 100*$x*$ and -180, represent the revenue generated from sales and the fixed costs associated with the business, respectively.

Understanding the Components

To truly appreciate the equation, let's dissect its components:

• -6$x^2$: This term represents the effect of price elasticity on profit. As the price increases, the quantity sold typically decreases, leading to a diminishing return on profit. The negative sign indicates that this effect becomes more pronounced at higher prices.
• 100$x$: This term signifies the revenue generated from selling soccer balls. It's a linear relationship, meaning that for each additional dollar in selling price, the revenue increases by $100, assuming the quantity sold remains constant.
• -180: This constant term represents the fixed costs associated with the business. These costs, such as rent, utilities, and salaries, remain constant regardless of the number of soccer balls sold.

The Significance of Zeroes

The zeroes of the function are the values of *$x*$ for which $y$ equals zero. In the context of our business, these zeroes represent the selling prices at which the daily profit is zero. These are the break-even points, where the business neither makes a profit nor incurs a loss. Identifying these points is crucial for setting a pricing strategy that ensures profitability.

Unveiling the Zeroes: A Step-by-Step Approach

To find the zeroes of the function $y = -6x^2 + 100x - 180$, we need to solve the quadratic equation $-6x^2 + 100x - 180 = 0$. There are several methods to solve quadratic equations, including factoring, completing the square, and the quadratic formula. In this case, the quadratic formula is the most efficient method.

The Quadratic Formula: A Reliable Tool

The quadratic formula is a universal solution for finding the roots of any quadratic equation in the form $ax^2 + bx + c = 0$. The formula is given by:

x = rac{-b ext{±} ext{√(}b^2 - 4ac)}{2a}

In our equation, $a = -6$, $b = 100$, and $c = -180$. Plugging these values into the quadratic formula, we get:

x = rac{-100 ext{±} ext{√(}100^2 - 4(-6)(-180))}{2(-6)}

Navigating the Calculations

Let's break down the calculations step by step:

1. Calculate the discriminant: $b^2 - 4ac = 100^2 - 4(-6)(-180) = 10000 - 4320 = 5680$

2. Take the square root of the discriminant: √5680 ≈ 75.37

3. Apply the quadratic formula:

   • x_1 = rac{-100 + 75.37}{-12} ≈ rac{-24.63}{-12} ≈ 2.05
   • x_2 = rac{-100 - 75.37}{-12} ≈ rac{-175.37}{-12} ≈ 14.61

Therefore, the zeroes of the function, rounded to the nearest hundredth, are approximately 2.05 and 14.61.

Interpreting the Zeroes: Break-Even Points and Beyond

The zeroes we've calculated, 2.05 and 14.61, represent the selling prices at which the daily profit is zero. These are the critical break-even points for the soccer ball business.

The Significance of Break-Even Points

• Lower Break-Even Point ($2.05): This is the minimum selling price required to cover all costs. Selling below this price will result in a loss.
• Upper Break-Even Point ($14.61): This is the maximum selling price at which the business can still break even. Selling above this price will also result in a loss, due to the decreasing demand for soccer balls at higher prices.

Charting the Path to Profitability

Knowing the break-even points is crucial, but it's only the first step. To maximize profit, we need to understand the entire profit curve. Since the parabola opens downwards, the maximum profit occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:

x_{vertex} = rac{-b}{2a}

In our case, x_{vertex} = rac{-100}{2(-6)} ≈ 8.33. This suggests that the selling price that maximizes profit is approximately $8.33.

The Importance of Market Dynamics

While the mathematical model provides valuable insights, it's essential to consider real-world market dynamics. Factors such as competition, demand, and consumer preferences can significantly impact the optimal selling price. A thorough market analysis should complement the mathematical analysis to arrive at a well-informed pricing strategy.

Conclusion: A Holistic Approach to Profit Optimization

In conclusion, understanding the zeroes of the profit function $y = -6x^2 + 100x - 180$ is a critical step in charting a path to profitability for our soccer ball business. The break-even points, calculated to be approximately $2.05 and $14.61, provide a crucial benchmark for pricing decisions. However, to truly optimize profit, we must consider the entire profit curve, identify the price that maximizes profit (approximately $8.33), and factor in real-world market dynamics.

This holistic approach, combining mathematical analysis with market understanding, will empower entrepreneurs to make informed decisions and navigate the complexities of the business world with confidence.