Two Companies Start Up At The Same Time—Company A Claims The

Two Companies Start Up At The Same Time Company A Claims Their Annual

Two companies start up at the same time. Company A claims their annual profits follow a linear model, P(x) = 10x − 7 where x is the number of units sold and x ≥ 1. Company B claims that their annual profits follow a radical model, P(x)=15 √ (x − 1) + 3 or P(x) = 15sqrt(x − 1) + 3, where x is the number of units sold and x ≥ 1. It is your job to investigate the validity of each claim. Choose five values for x to plug into the linear function, P(x) = 10x − 7 and create a table of values. Use the same five x values to plug into the radical function, P(x)=15 √ x − 1 + 3 or P(x) = 15sqrt(x − 1) + 3, and create a table of values. Using the table of values from parts 1 and 2, graph both functions. Compare the profits of each company based on the graphs and evaluate their claims. Which model seems more realistic, the linear or radical model, and why? Of the function types studied—linear, quadratic, radical, or rational—which is more likely to represent a real profit function for a company? Provide a scenario for a specific type of company and justify your choice.

Paper For Above instruction

The analysis of profit models for companies is fundamental in understanding their economic behavior and predicting future performance. This paper investigates two competing profit models—linear and radical—for two startup companies, examining their validity through calculated data and graphical comparison, then assessing which model offers a more realistic representation of business profits in a real-world context.

The linear model, P(x) = 10x - 7, posits a consistent profit increase per unit sold, typical of companies with stable production costs and proportional revenue streams. To analyze this, select five values of x (units sold): for instance, x = 1, 2, 3, 4, 5. Calculating P(x) for these values yields:

• x = 1: P(1) = 10(1) - 7 = 3
• x = 2: P(2) = 20 - 7 = 13
• x = 3: P(3) = 30 - 7 = 23
• x = 4: P(4) = 40 - 7 = 33
• x = 5: P(5) = 50 - 7 = 43

Correspondingly, for the radical model P(x) = 15√(x - 1) + 3, the same x values produce:

• x = 1: P(1) = 15√(0) + 3 = 3
• x = 2: P(2) = 15√(1) + 3 ≈ 15(1) + 3 = 18
• x = 3: P(3) = 15√(2) + 3 ≈ 15(1.414) + 3 ≈ 21.21 + 3 ≈ 24.21
• x = 4: P(4) = 15√(3) + 3 ≈ 15(1.732) + 3 ≈ 25.98 + 3 ≈ 28.98
• x = 5: P(5) = 15√(4) + 3 = 15(2) + 3 = 30 + 3 = 33

Plotting these points on a graph reveals that the linear model depicts a steady, consistent increase in profit with each additional unit sold, while the radical model suggests a rapid initial increase that slows as sales grow, eventually approaching a maximum limit.

The graph indicates that for small values of x, the radical model predicts higher profits than the linear model, but as sales increase, the two models converge. When analyzing the plausibility of these models, the linear model’s assumption of a constant profit per unit is suitable for companies with fixed costs and revenues that grow proportionally. Conversely, the radical model might represent companies experiencing diminishing returns or saturation effects—such as tech startups saturating a market or products with limited scalability. For instance, a company providing a service with rapid initial gains—like a viral app—could initially experience high profit jumps that taper off as market saturation occurs.

Evaluating these models in real-world scenarios, the linear model offers simplicity and steady growth, often applicable to manufacturing firms where costs and revenues are fixed per unit. The radical model better captures the nonlinear nature of many real-world phenomena where growth accelerates initially but slows over time due to market saturation, diminishing marginal returns, or resource limits. For example, the profits of a renewable energy company may follow a radical pattern as initial investments yield substantial returns, which slow as the market approaches saturation.

Among the function types—linear, quadratic, radical, and rational—the quadratic function tends to best represent profit in cases involving increasing returns up to a point followed by diminishing returns or losses, such as in industries with startup costs and scaling. Rational functions, which incorporate divisions by variables, often depict scenarios with thresholds or limits, such as when profits suffer due to bottlenecks or maximum capacity constraints. However, the most common and straightforward for many companies remains the linear function for steady growth and the quadratic or radical functions for growth with eventual stabilization.

In conclusion, the choice of profit model depends on the nature of the company and its market. A small manufacturing firm may exhibit linear profit growth, while a tech startup experiencing rapid initial growth followed by saturation aligns better with a radical (square root) model. Recognizing the underlying economic behaviors and market limitations helps in selecting the most accurate and useful profit models for planning and forecasting.

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