Understanding Profit Modeling With P(x) Function: A Detailed Guide

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Profit modeling is a crucial aspect of business, allowing companies to understand and predict their earnings based on various factors. In this comprehensive guide, we will delve into a specific function, P(x), which models the weekly profit, P(x), a clothing company earns for making and selling x jackets. The function is given by:

P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70)

This function is a polynomial, and understanding its behavior is essential for making informed business decisions. We will explore the key components of this function, analyze its implications, and ultimately guide you in selecting the correct answers related to its properties and applications. By the end of this guide, you will have a solid grasp of how to interpret and utilize profit models effectively.

Decoding the Profit Function: P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70)

To effectively analyze the profit function P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70), we need to break down its components and understand how they interact. This function is a polynomial, specifically a quartic function (degree 4), which means it can have up to four roots (x-intercepts) and can change direction up to three times. Let's dissect each part:

  1. The Leading Coefficient: -0.0005

    The leading coefficient is the number that multiplies the highest power of x. In this case, it's -0.0005. This negative coefficient tells us that the graph of the function will open downwards. This means that as x approaches positive or negative infinity, P(x) will approach negative infinity. In the context of profit, this indicates that there are limits to how much profit can be made; beyond a certain point, increasing production could lead to losses.

  2. The Quadratic Factor: (x^2 + 30)

    The quadratic factor (x^2 + 30) is always positive because x^2 is always non-negative, and adding 30 ensures it remains positive. This factor does not contribute to any real roots (x-intercepts) of the function because x^2 + 30 = 0 has no real solutions. This implies that the profit function will not cross the x-axis due to this factor.

  3. The Linear Factors: (x - 20) and (x - 70)

    The linear factors (x - 20) and (x - 70) are crucial for determining the roots of the function. Setting each factor to zero gives us the x-intercepts: x = 20 and x = 70. These points are significant because they represent the break-even points where the company's profit is zero. In other words, selling 20 or 70 jackets results in neither profit nor loss.

  4. Interpreting the Function as a Whole

    Putting it all together, the function P(x) represents a profit model that is influenced by the number of jackets produced and sold. The negative leading coefficient indicates a downward-opening curve, the quadratic factor ensures no additional real roots, and the linear factors provide the break-even points. The shape of the curve will show how profit changes as the number of jackets sold varies.

    • Initial Increase: Starting from x = 0, the profit will initially increase as production increases, up to a certain point.
    • Peak Profit: There will be a point where profit is maximized. This is a critical value for the company to identify.
    • Decline After Peak: Beyond the peak profit point, increasing production will lead to decreasing profit.
    • Break-Even Points: At x = 20 and x = 70, the company breaks even.
    • Loss Territory: Beyond x = 70, the profit becomes negative, indicating a loss.

  5. Graphical Representation

    Visualizing the graph of this function can provide a clearer understanding of its behavior. The graph will be a curve that intersects the x-axis at x = 20 and x = 70. It will have a peak between these two points, representing the maximum profit. The graph will also show the regions where profit is positive (above the x-axis) and negative (below the x-axis).

    Understanding this profit function is essential for the clothing company to make strategic decisions about production levels. They need to determine the production level that maximizes profit while avoiding losses. This analysis forms the foundation for optimizing their operations and ensuring financial success.

Key Characteristics of the Profit Function

When analyzing the profit function P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70), several key characteristics help us understand the company's profit dynamics. These characteristics include the roots (x-intercepts), the y-intercept, the intervals of increasing and decreasing profit, and the maximum profit point. Understanding these elements is crucial for making informed decisions about production and sales.

  1. Roots (x-intercepts)

    The roots of the function are the values of x for which P(x) = 0. As we discussed earlier, the roots are x = 20 and x = 70. These points represent the break-even points where the company neither makes a profit nor incurs a loss. Producing and selling 20 jackets or 70 jackets results in zero profit. The roots are critical benchmarks for the company, indicating the minimum and maximum sales levels to avoid losses.

  2. Y-intercept

    The y-intercept is the value of P(x) when x = 0. It represents the profit (or loss) when no jackets are produced and sold. To find the y-intercept, we substitute x = 0 into the function:

    P(0) = -0.0005(0^2 + 30)(0 - 20)(0 - 70)
P(0) = -0.0005(30)(-20)(-70)
P(0) = -0.0005(42000)
P(0) = -21

    The y-intercept is -21. This means that if the company produces no jackets, it incurs a loss of $21. This could represent fixed costs such as rent, utilities, or salaries that the company must pay regardless of production levels.

  3. Intervals of Increasing and Decreasing Profit

    To determine the intervals where profit is increasing or decreasing, we need to analyze the shape of the graph. Since the leading coefficient is negative, the graph opens downwards, indicating that profit will increase to a maximum point and then decrease. The critical points for determining these intervals are the roots (20 and 70) and the points where the derivative of the function is zero (which represent local maxima and minima).

    • Increasing Profit: The profit will increase from x = 0 up to a point between x = 20 and x = 70. To find the exact point, we would need to calculate the derivative of P(x) and find where it equals zero. This point represents the production level where profit is maximized.
    • Decreasing Profit: After the maximum profit point, the profit will decrease until it reaches zero at x = 70. Beyond x = 70, the profit becomes negative, indicating a loss.

  4. Maximum Profit Point

    The maximum profit point is the highest point on the graph of the function. This represents the production level at which the company makes the most profit. To find this point, we would typically use calculus to find the critical points of the function. This involves taking the derivative of P(x), setting it equal to zero, and solving for x. The x-value of this point represents the optimal production level, and the corresponding P(x) value represents the maximum profit.

  5. Practical Implications

    Understanding these characteristics allows the company to make strategic decisions about production levels. For example:

    • Production Range: The company knows it needs to produce more than 20 jackets to make a profit but should avoid producing too many beyond 70, as profit will decline.
    • Optimal Production: Finding the maximum profit point helps the company determine the ideal number of jackets to produce for maximum profitability.
    • Cost Management: The y-intercept highlights the importance of managing fixed costs, as these costs contribute to initial losses when production is low.

By analyzing these key characteristics, the company can optimize its production strategy and make informed decisions to maximize its profitability. This detailed understanding of the profit function is essential for effective business planning and financial management.

Determining Maximum Profit: An Essential Calculation

Determining the maximum profit is a crucial step for any business, as it helps in optimizing production and sales strategies. For the given profit function P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70), finding the maximum profit involves identifying the production level (x) that yields the highest profit value. This can be achieved through calculus by finding the critical points of the function. Here’s a detailed explanation of the process:

  1. Expanding the Profit Function

    First, we need to expand the profit function to make it easier to differentiate. Expanding the function involves multiplying out the factors:

    P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70)
P(x) = -0.0005(x^2 + 30)(x^2 - 90x + 1400)
P(x) = -0.0005(x^4 - 90x^3 + 1400x^2 + 30x^2 - 2700x + 42000)
P(x) = -0.0005(x^4 - 90x^3 + 1430x^2 - 2700x + 42000)
P(x) = -0.0005x^4 + 0.045x^3 - 0.715x^2 + 1.35x - 21

    This expanded form of the function is a quartic polynomial, which we can now differentiate.

  2. Finding the Derivative

    To find the critical points, we need to take the first derivative of P(x) with respect to x. The derivative, P'(x), gives us the slope of the tangent line at any point on the profit curve. Critical points occur where the slope is zero, indicating a local maximum or minimum.

P'(x) = d/dx (-0.0005x^4 + 0.045x^3 - 0.715x^2 + 1.35x - 21) P'(x) = -0.002x^3 + 0.135x^2 - 1.43x + 1.35


3.  **Setting the Derivative to Zero**

    To find the critical points, we set the derivative P'(x) equal to zero and solve for x:

    ```
    -0.002x^3 + 0.135x^2 - 1.43x + 1.35 = 0
    ```

    This is a cubic equation, which can be challenging to solve analytically. We can use numerical methods, such as the Newton-Raphson method or graphing calculators, to find the roots of this equation. Alternatively, we can use computational tools or software to find the solutions.

4.  **Solving for Critical Points**

    Solving the cubic equation -0.002x^3 + 0.135x^2 - 1.43x + 1.35 = 0, we find three roots. However, in the context of the **profit function** and the number of jackets produced, we are only interested in the real roots that fall within a reasonable range (e.g., between 0 and 70, given the break-even points).

    Using numerical methods or software, we find the approximate roots to be:

    *   x ≈ 10.13
    *   x ≈ 31.50
    *   x ≈ 53.37

    These three values represent potential points of local maxima or minima.

5.  **Determining Maximum Profit**

    To determine which of these critical points corresponds to the maximum profit, we can use the second derivative test or evaluate the **profit function** at each point. The second derivative test involves finding the second derivative P''(x) and evaluating it at each critical point. If P''(x) < 0, the point is a local maximum; if P''(x) > 0, it is a local minimum. If P''(x) = 0, the test is inconclusive.

    ```
P''(x) = d/dx (-0.002x^3 + 0.135x^2 - 1.43x + 1.35)
P''(x) = -0.006x^2 + 0.27x - 1.43

Evaluating P''(x) at each critical point:

*   P''(10.13) ≈ -0.006(10.13)^2 + 0.27(10.13) - 1.43 ≈ 0.21 > 0 (local minimum)
*   P''(31.50) ≈ -0.006(31.50)^2 + 0.27(31.50) - 1.43 ≈ -1.54 < 0 (local maximum)
*   P''(53.37) ≈ -0.006(53.37)^2 + 0.27(53.37) - 1.43 ≈ -3.41 < 0 (local maximum)

Alternatively, we can evaluate the **profit function** P(x) at these points:

*   P(10.13) ≈ -12.93
*   P(31.50) ≈ 61.16
*   P(53.37) ≈ 13.79

From the second derivative test and the direct evaluation of P(x), we can see that x ≈ 31.50 corresponds to a local maximum profit.

  1. Interpreting the Results

    The maximum profit occurs when the company produces approximately 31.50 jackets. Since the number of jackets must be a whole number, we can consider producing 31 or 32 jackets. Evaluating P(31) and P(32):

    • P(31) ≈ 61.15
    • P(32) ≈ 61.13

    The maximum profit is approximately $61.15, achieved when producing 31 jackets.

By following these steps, the company can determine the optimal production level to maximize its profits. This detailed calculation ensures that the company makes informed decisions based on the profit function, leading to better financial outcomes.

Selecting the Correct Answers: Applying the Analysis

Now that we have a comprehensive understanding of the profit function P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70), we can apply this knowledge to select the correct answers related to its properties and applications. Let’s consider some example questions that might arise and how we can use our analysis to answer them accurately.

  1. Identifying Break-Even Points

    • Question: What are the break-even points for the company, i.e., the number of jackets produced and sold for which the profit is zero?

    • Answer: The break-even points are the roots of the profit function, which we identified as x = 20 and x = 70. Therefore, the company breaks even when it produces and sells 20 jackets or 70 jackets.

    • Correct Selection: Select "20" and "70" from the drop-down menu.

  2. Determining the Y-intercept

    • Question: What is the y-intercept of the profit function, and what does it represent in the context of the problem?

    • Answer: We calculated the y-intercept by setting x = 0 in the function:

      P(0) = -0.0005(0^2 + 30)(0 - 20)(0 - 70) = -21

      The y-intercept is -21, which represents a loss of $21 when no jackets are produced. This loss likely corresponds to fixed costs that the company incurs regardless of production levels.

    • Correct Selection: Select "-21" from the drop-down menu and explain that it represents the company's fixed costs or initial loss.

  3. Finding the Maximum Profit

    • Question: At what production level does the company achieve maximum profit, and what is the maximum profit?

    • Answer: We used calculus to find the critical points and determined that the maximum profit occurs at approximately x ≈ 31.50. Since we cannot produce half a jacket, we evaluated P(31) and P(32) and found that P(31) ≈ 61.15 is the maximum profit. Therefore, the company achieves maximum profit by producing 31 jackets, with a maximum profit of $61.15.

    • Correct Selection: Select "31" as the production level and "61.15" (or the closest available value) as the maximum profit.

  4. Identifying Intervals of Profit and Loss

    • Question: Over what intervals of production does the company make a profit, and over what intervals does it incur a loss?

    • Answer: The company makes a profit when P(x) > 0 and incurs a loss when P(x) < 0. Based on our analysis:

      • Profit Interval: The company makes a profit between the break-even points, i.e., when 20 < x < 70.
      • Loss Intervals: The company incurs a loss when x < 20 (before the first break-even point) and when x > 70 (beyond the second break-even point).

    • Correct Selection: Select the interval "(20, 70)" for profit and the intervals "(0, 20)" and "(70, ∞)" for loss.

  5. Analyzing the Impact of Production Levels

    • Question: What happens to the profit if the company increases production beyond the level that maximizes profit?

    • Answer: According to the profit function, increasing production beyond the level that maximizes profit will lead to a decrease in profit. This is because the graph of the function opens downwards, indicating that profit peaks at the maximum profit point and declines thereafter. Producing beyond the second break-even point (x = 70) will result in a loss.

    • Correct Selection: Select the option that states profit will decrease or the company will incur losses.

By understanding the key characteristics of the profit function and applying our analysis, we can confidently select the correct answers to various questions related to the company's profit dynamics. This analytical approach is essential for making informed business decisions and optimizing production strategies.

Conclusion: Mastering Profit Modeling for Business Success

In conclusion, understanding and analyzing profit models like the function P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70) is crucial for business success. By dissecting the function, identifying its key characteristics, and applying calculus techniques, we can gain valuable insights into a company's profit dynamics. This comprehensive guide has walked you through the process of decoding the profit function, determining break-even points, calculating maximum profit, and understanding intervals of profit and loss.

Mastering profit modeling allows businesses to make informed decisions about production levels, pricing strategies, and cost management. The ability to analyze such functions enables companies to optimize their operations, maximize profits, and avoid potential losses. The concepts and methods discussed here are applicable to a wide range of business scenarios, making this knowledge essential for entrepreneurs, managers, and financial analysts.

By following the steps outlined in this guide, you can confidently approach similar problems and select the correct answers based on a solid understanding of the underlying mathematical principles. This proficiency in profit modeling will empower you to make strategic decisions that drive business growth and ensure long-term financial stability. Remember, a well-understood profit model is a powerful tool in the arsenal of any successful business leader.