Calculating Profit For Home-Based Sign Company - A Detailed Guide

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This article delves into the critical aspect of profit maximization for a home-based sign company. Profit maximization is a cornerstone of any successful business, and understanding the factors that influence it is paramount. For a sign company, the price at which each sign is sold plays a pivotal role in determining the overall profit. We will explore how a mathematical function can model the relationship between the price of a sign and the company's monthly profit, enabling informed decision-making to optimize financial performance.

Understanding the Profit Function

The profit function provided, p(x)=−10x2+498x−1500p(x) = -10x^2 + 498x - 1500, is a quadratic equation. This type of equation creates a parabolic curve when graphed, which is essential for understanding how the profit changes as the price (xx) varies. The negative coefficient in front of the x2x^2 term indicates that the parabola opens downwards, meaning there is a maximum point, representing the price that yields the highest profit. The other terms in the equation, 498x498x and −1500-1500, represent the revenue generated from selling the signs and the fixed costs associated with running the business, respectively.

Key Components of the Profit Function

  • −10x2-10x^2: This term represents the quadratic effect of price on profit. As the price increases, the revenue initially increases, but eventually, the effect diminishes due to the squared term. This reflects the principle of diminishing returns, where at some point, increasing the price may lead to fewer sales, thus impacting overall profit negatively.
  • 498x498x: This term represents the linear relationship between price and revenue. For each sign sold at a price xx, the company earns 498x498x. This signifies the direct income generated from sales.
  • −1500-1500: This constant term represents the fixed costs of the business. These costs, such as rent, utilities, and material expenses, remain constant regardless of the number of signs sold. They represent a baseline expense that the company needs to cover.

Evaluating Profit at a Specific Price

The question asks for the company's profit if each sign is sold for $20. To determine this, we substitute x=20x = 20 into the profit function: p(20)=−10(20)2+498(20)−1500p(20) = -10(20)^2 + 498(20) - 1500. Evaluating this expression will give us the profit earned when the sign price is set at $20.

Calculating Profit for Signs Sold at $20

To calculate the profit when the company sells each sign for $20, we need to substitute x=20x = 20 into the given profit function: p(x)=−10x2+498x−1500p(x) = -10x^2 + 498x - 1500. This involves careful arithmetic and a clear understanding of the order of operations. By accurately substituting and simplifying the equation, we can determine the profit earned at this particular price point. This calculation is a crucial step in understanding the financial implications of pricing decisions for the business.

Step-by-Step Calculation

  1. Substitute x=20x = 20 into the equation: p(20)=−10(20)2+498(20)−1500p(20) = -10(20)^2 + 498(20) - 1500
  2. Calculate the square: p(20)=−10(400)+498(20)−1500p(20) = -10(400) + 498(20) - 1500
  3. Perform the multiplications: p(20)=−4000+9960−1500p(20) = -4000 + 9960 - 1500
  4. Add and subtract the values: p(20)=4460p(20) = 4460

Therefore, when the company sells each sign for $20, the profit is $4460. This result highlights the importance of understanding the profit function and how it can be used to calculate profitability at different price points. Analyzing such calculations can help the company make informed decisions about pricing strategies.

The Significance of the Profit Calculation

This calculation is not just about finding a numerical answer; it's about understanding the financial dynamics of the business. The calculated profit of $4460 when selling signs for $20 provides valuable insights into the company's operational efficiency and pricing strategy. It serves as a benchmark against which the company can evaluate other potential pricing points and business decisions. Furthermore, it allows the company to project future earnings and make necessary adjustments to their strategy to optimize profitability. This proactive approach to financial management is essential for sustainable growth and success in a competitive market.

Analyzing the Result

  • The profit of $4460 suggests that selling signs at $20 is a viable option.
  • However, it's crucial to analyze other price points to determine if this is the optimal price for profit maximization.
  • The company should consider factors like market demand, competition, and cost of production when making pricing decisions.

Determining the Optimal Price for Maximum Profit

To determine the optimal price that maximizes profit, we need to find the vertex of the parabola represented by the profit function. The vertex is the highest point on the parabola, corresponding to the maximum profit. The x-coordinate of the vertex represents the price that yields the maximum profit. There are two primary methods to find the vertex: completing the square and using the vertex formula. Each method provides a reliable way to determine the optimal price point.

Method 1: Completing the Square

Completing the square involves rewriting the quadratic equation in vertex form, which is p(x)=a(x−h)2+kp(x) = a(x - h)^2 + k, where (h,k)(h, k) represents the vertex of the parabola. This method provides a clear visual representation of the profit function and allows for easy identification of the vertex. However, it can be a more complex method compared to using the vertex formula.

  1. Start with the profit function: p(x)=−10x2+498x−1500p(x) = -10x^2 + 498x - 1500
  2. Factor out the coefficient of the x2x^2 term from the first two terms: p(x)=−10(x2−49.8x)−1500p(x) = -10(x^2 - 49.8x) - 1500
  3. Complete the square inside the parenthesis: To do this, take half of the coefficient of the xx term (-49.8), square it, and add and subtract it inside the parenthesis. Half of -49.8 is -24.9, and (-24.9)^2 = 620.01. So, we have: p(x)=−10(x2−49.8x+620.01−620.01)−1500p(x) = -10(x^2 - 49.8x + 620.01 - 620.01) - 1500
  4. Rewrite the expression inside the parenthesis as a square: p(x)=−10((x−24.9)2−620.01)−1500p(x) = -10((x - 24.9)^2 - 620.01) - 1500
  5. Distribute the -10: p(x)=−10(x−24.9)2+6200.1−1500p(x) = -10(x - 24.9)^2 + 6200.1 - 1500
  6. Simplify: p(x)=−10(x−24.9)2+4700.1p(x) = -10(x - 24.9)^2 + 4700.1

From the vertex form, we can see that the vertex is (24.9,4700.1)(24.9, 4700.1). This means that the price that maximizes profit is approximately $24.9, and the maximum profit is $4700.1.

Method 2: Using the Vertex Formula

The vertex formula provides a direct way to find the x-coordinate (h) of the vertex. For a quadratic equation in the form p(x)=ax2+bx+cp(x) = ax^2 + bx + c, the x-coordinate of the vertex is given by h=−b/(2a)h = -b / (2a). This method is generally quicker and easier than completing the square, making it a popular choice for finding the optimal price.

  1. Identify the coefficients in the profit function: a=−10a = -10, b=498b = 498, and c=−1500c = -1500
  2. Apply the vertex formula: h=−498/(2∗−10)=−498/−20=24.9h = -498 / (2 * -10) = -498 / -20 = 24.9

So, the price that maximizes profit is $24.9. To find the maximum profit, substitute this value back into the profit function:

p(24.9)=−10(24.9)2+498(24.9)−1500p(24.9) = -10(24.9)^2 + 498(24.9) - 1500 p(24.9)=−10(620.01)+12300.2−1500p(24.9) = -10(620.01) + 12300.2 - 1500 p(24.9)=−6200.1+12400.2−1500p(24.9) = -6200.1 + 12400.2 - 1500 p(24.9)=4700.1p(24.9) = 4700.1

Thus, the maximum profit is $4700.1 when the price is set at $24.9.

Conclusion: Optimizing Pricing Strategy for Profitability

In conclusion, understanding the relationship between pricing and profit is crucial for a home-based sign company's success. By using the profit function p(x)=−10x2+498x−1500p(x) = -10x^2 + 498x - 1500, we can calculate profit at different price points and determine the optimal pricing strategy for profit maximization. The calculation shows that selling each sign for $20 results in a profit of $4460. However, to maximize profit, the company should set the price at approximately $24.9, which yields a maximum profit of $4700.1. This analysis underscores the importance of using mathematical models to inform business decisions and optimize financial performance. Continuously evaluating and adjusting pricing strategies based on market conditions and cost factors will help the company maintain profitability and achieve long-term success. This comprehensive approach ensures that the company can adapt to changing market dynamics and sustain its growth trajectory.

By analyzing the profit function and determining the optimal price, the company can make informed decisions to maximize its earnings and achieve sustainable growth. This analytical approach is essential for long-term success in a competitive market.

Answer to the Question

Based on our calculation, the company's profit if it sells each sign for $20 is $4460. Therefore, the correct answer is not among the provided options. It is crucial to perform calculations accurately to ensure the correct answer is obtained. This reinforces the importance of double-checking calculations and understanding the underlying principles to arrive at the right solution.

Key Takeaways

  • Understanding the Profit Function: The profit function is a crucial tool for analyzing the relationship between price and profit.
  • Calculating Profit: Substituting the price into the profit function allows for the calculation of profit at different price points.
  • Optimal Pricing Strategy: Determining the optimal price requires finding the vertex of the parabola, which can be done through completing the square or using the vertex formula.
  • Informed Decision-Making: Utilizing mathematical models and calculations can significantly improve business decision-making and financial performance.

By mastering these concepts, a home-based sign company can effectively manage its pricing strategy and maximize its profitability. This strategic approach is essential for sustainable growth and success in a dynamic market environment.