The Equation of Springtime Bouquet Profits

Oct 13, 2025 by talynt.org 22 views

Introduction

In this article, we will delve into the world of mathematics and explore the equation that models the profits from selling springtime bouquets. The equation, given as $P = -5b^2 + 450b - 1,000$ , where $b$ represents the number of springtime bouquets, is a quadratic equation that can be used to determine the maximum profit that Sarah's flower shop can achieve. In this discussion, we will analyze the given equation and determine which statement is true.

The Equation of Springtime Bouquet Profits

The equation $P = -5b^2 + 450b - 1,000$ represents the profits from selling springtime bouquets. To understand this equation, let's break it down into its individual components.

The coefficient of the squared term, $-5$ , represents the rate at which the profit decreases as the number of bouquets sold increases. A negative coefficient indicates that the profit decreases as the number of bouquets sold increases.
The coefficient of the linear term, $450$ , represents the rate at which the profit increases as the number of bouquets sold increases. A positive coefficient indicates that the profit increases as the number of bouquets sold increases.
The constant term, $-1,000$ , represents the initial profit or loss that the flower shop incurs when no bouquets are sold.

Analyzing the Equation

To determine which statement is true, we need to analyze the equation and its components. Let's start by finding the vertex of the parabola represented by the equation. The vertex of a parabola is the point at which the parabola changes direction, and it is represented by the equation $x = -\frac{b}{2a}$ .

In this case, the equation is $P = -5b^2 + 450b - 1,000$ , so we can plug in the values of $a$ and $b$ to find the vertex.

$a = -5$
$b = 450$

Plugging these values into the equation, we get:

$x = -\frac{450}{2(-5)}$ $x = -\frac{450}{-10}$ $x = 45$

So, the vertex of the parabola is at $x = 45$ . This means that the maximum profit that Sarah's flower shop can achieve is when $45$ bouquets are sold.

Determining the Correct Statement

Now that we have analyzed the equation and found the vertex, we can determine which statement is true. Let's consider the following statements:

Statement 1: The profit increases as the number of bouquets sold increases.
Statement 2: The profit decreases as the number of bouquets sold increases.
Statement 3: The maximum profit is achieved when $45$ bouquets are sold.

To determine which statement is true, we need to analyze the equation and its components. Let's start by examining the coefficient of the linear term, $450$ . Since the coefficient is positive, it indicates that the profit increases as the number of bouquets sold increases.

However, we also need to consider the coefficient of the squared term, $-5$ . Since the coefficient is negative, it indicates that the profit decreases as the number of bouquets sold increases. This means that the profit increases at a decreasing rate as the number of bouquets sold increases.

Now, let's examine the vertex of the parabola, which is at $x = 45$ . This means that the maximum profit is achieved when $45$ bouquets are sold.

Based on our analysis, we can conclude that the correct statement is:

Statement 3: The maximum profit is achieved when $45$ bouquets are sold.

Conclusion

In this article, we analyzed the equation that models the profits from selling springtime bouquets. We broke down the equation into its individual components and analyzed the coefficient of the linear term, the coefficient of the squared term, and the constant term. We also found the vertex of the parabola, which represents the maximum profit that Sarah's flower shop can achieve.

Based on our analysis, we determined that the correct statement is:

Statement 3: The maximum profit is achieved when $45$ bouquets are sold.

This conclusion is supported by the fact that the coefficient of the linear term is positive, indicating that the profit increases as the number of bouquets sold increases. However, the coefficient of the squared term is negative, indicating that the profit decreases as the number of bouquets sold increases. The vertex of the parabola, which is at $x = 45$ , represents the maximum profit that Sarah's flower shop can achieve.

Final Answer

The final answer is:

Statement 3: The maximum profit is achieved when $45$ bouquets are sold.
Understanding the Equation of Springtime Bouquet Profits: Q&A ===========================================================

Introduction

In our previous article, we analyzed the equation that models the profits from selling springtime bouquets. We broke down the equation into its individual components and determined that the maximum profit is achieved when $45$ bouquets are sold. In this article, we will answer some frequently asked questions about the equation and its components.

Q&A

Q: What is the equation that models the profits from selling springtime bouquets?

A: The equation that models the profits from selling springtime bouquets is $P = -5b^2 + 450b - 1,000$ , where $b$ represents the number of springtime bouquets.

Q: What is the coefficient of the squared term in the equation?

A: The coefficient of the squared term in the equation is $-5$ . This means that the profit decreases as the number of bouquets sold increases.

Q: What is the coefficient of the linear term in the equation?

A: The coefficient of the linear term in the equation is $450$ . This means that the profit increases as the number of bouquets sold increases.

Q: What is the constant term in the equation?

A: The constant term in the equation is $-1,000$ . This means that the flower shop incurs an initial loss of $1,000$ when no bouquets are sold.

Q: Where is the vertex of the parabola represented by the equation?

A: The vertex of the parabola represented by the equation is at $x = 45$ . This means that the maximum profit is achieved when $45$ bouquets are sold.

Q: How does the profit change as the number of bouquets sold increases?

A: The profit increases at a decreasing rate as the number of bouquets sold increases. This means that the profit increases initially, but at a decreasing rate.

Q: What is the maximum profit that Sarah's flower shop can achieve?

A: The maximum profit that Sarah's flower shop can achieve is when $45$ bouquets are sold.

Q: How can the equation be used in real-world scenarios?

A: The equation can be used to determine the maximum profit that a flower shop can achieve by selling springtime bouquets. It can also be used to determine the optimal number of bouquets to sell in order to maximize profits.

Conclusion

In this article, we answered some frequently asked questions about the equation that models the profits from selling springtime bouquets. We broke down the equation into its individual components and determined that the maximum profit is achieved when $45$ bouquets are sold. We also discussed how the profit changes as the number of bouquets sold increases and how the equation can be used in real-world scenarios.

Final Answer

The final answer is:

The maximum profit is achieved when $45$ bouquets are sold.

Additional Resources

For more information on the equation that models the profits from selling springtime bouquets, please refer to the following resources:

Disclaimer

The information provided in this article is for educational purposes only. It is not intended to be used as a substitute for professional advice or guidance. If you are a flower shop owner or manager, please consult with a qualified professional before making any decisions based on the information provided in this article.