Ma 218 Classwork 23.1: Rafael Estimates It Costs $14

Ma 218 Classwork 23 1 Rafael Estimates That It Costs 14 To Produ

Ma 218 Classwork 23 1 Rafael estimates the costs, revenue, and profit functions for a calculator business, along with analyzing profit relationships for fast-food items. The assignment involves creating mathematical functions, graphing these functions, interpreting their intersections, and applying algebraic techniques to determine profit scenarios and strategic decisions.

Paper For Above instruction

Introduction

Understanding the relationship between costs, revenues, and profits is fundamental in business analysis. This essay explores the mathematical functions that model these relationships for Rafael’s calculator production and McDonald’s fast-food sales, analyzing their implications for profitability and strategic business decisions.

1. Cost, Revenue, and Profit Functions for Rafael’s Calculator Business

Rafael estimates that it costs $14 to produce each calculator, and the selling price per calculator is $23. Additionally, the factory incurs fixed costs of $1200 regardless of the number of calculators produced. To mathematically model this scenario:

  • Cost function C(q): This function represents the total cost of producing q calculators. Since variable costs are $14 per calculator and fixed costs are $1200, the total cost is the sum of fixed and variable costs.

Mathematically, this is expressed as:

C(q) = 14q + 1200

This function explains how total costs increase linearly with the number of calculators manufactured.

  • Revenue function R(q): Revenue depends on the number of calculators sold and the selling price per calculator. With each calculator priced at $23, the total revenue for q calculators sold is:
R(q) = 23q

This function demonstrates the linear relationship between the quantity sold and total income generated.

  • Profit function P(q): Profit is the difference between revenue and costs. Combining the previous functions yields:
P(q) = R(q) - C(q) = 23q - (14q + 1200) = 9q - 1200

This function helps analyze profitability at various levels of production and sales.

2. Graphical Analysis and Intersection of Functions

Graphing all three functions on the same coordinate plane provides visual insight into their relationships. The cost function (C(q)) is a straight line starting at $1200 when no calculators are produced, with a slope of 14. The revenue function (R(q)) starts at zero and rises with a slope of 23. The profit function (P(q)) is a line with slope 9, starting at -$1200 when no calculators are sold.

The intersection point of R(q) and C(q) is critical — it indicates the break-even point where total revenue exactly covers total costs, meaning no profit or loss. Solving for q when R(q) = C(q):

23q = 14q + 1200

9q = 1200


q = 133.33

Thus, selling approximately 134 calculators yields zero profit; sales beyond this point generate profit, while fewer result in losses.

3. Profit Analysis for Specific Sales Numbers

When 100 calculators are sold, the profit is:

P(100) = 9(100) - 1200 = 900 - 1200 = -$300

indicating a loss of $300. When 2000 calculators are sold:

P(2000) = 9(2000) - 1200 = 18,000 - 1200 = $16,800

representing a substantial profit. This illustrates the importance of sales volume in achieving profitability.

4. Profit Functions for McDonald's Fast Food Items

The report states profit per item: 39 cents for fries, 96 cents for drinks, and a loss of 6 cents for burgers. Assign variables: x = number of fries, y = number of drinks, z = number of burgers. The profit function in dollars is:

Profit = 0.39x + 0.96y - 0.06z

This formula models the combined profit from the three items based on their quantities.

5. Adjusted Daily Profit with Fixed Burger Sales

If Bob’s McDonald’s sells 500 burgers daily, then z = 500, and profit becomes:

Profit = 0.39x + 0.96y - 0.06(500) = 0.39x + 0.96y - 30

This simplifies profit calculation considering fixed burger sales.

6. Minimum Drinks Needed for Target Daily Profit

Bob wants at least $650 daily profit. Substituting z = 500, the inequality is:

0.39x + 0.96y - 30 ≥ 650

0.39x + 0.96y ≥ 680

To find the minimum y (drinks), considering x (fries):

0.96y ≥ 680 - 0.39x

y ≥ (680 - 0.39x) / 0.96

This inequality demonstrates how many drinks must be sold depending on the fries sold to ensure profitable daily operations.

7. Graphical Representation and Profitability Region

Plotting the inequality y ≥ (680 - 0.39x) / 0.96 in a coordinate system with x (fries) on the horizontal axis and y (drinks) on the vertical axis depicts the profitable region. The boundary line represents the minimum y for given x. The shaded area above this line indicates combinations of fries and drinks yielding at least $650 profit daily, guiding strategic sales plans.

8. Strategic Business Implications

Major fast-food chains like McDonald’s often sell items at loss to attract customers, increase traffic, and promote sales of higher-margin items. Loss-leaders can stimulate overall store profitability and customer loyalty. For franchise owners, analyzing profit functions allows for optimal menu and sales strategies—for example, focusing on high-margin items while balancing the volume of low-margin or loss-leading products like hamburgers. Using these models quantitatively helps in identifying the most profitable sales mix and setting targets to maximize revenue and profit margins.

Conclusion

Modeling business operations with cost, revenue, and profit functions provides invaluable insights into operational efficiency and strategic decision-making. Graphical analysis of these functions reveals critical points like break-even sales volumes and profit thresholds. Furthermore, algebraic inequalities help determine necessary sales targets to achieve desired profitability levels. Such mathematical tools are essential for business owners aiming to optimize their sales strategies, control costs, and enhance overall profitability in competitive markets.

References

  • Baron, J. (2010). Managerial Economics and Business Strategy. McGraw-Hill Education.
  • Frank, R. H., & Bernanke, B. S. (2007). Principles of Economics. McGraw-Hill/Irwin.
  • Hirschey, M. (2014). Financial Fundamentals. South-Western College Pub.
  • Mankiw, N. G. (2014). Principles of Economics. Cengage Learning.
  • Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2013). Fundamentals of Corporate Finance. McGraw-Hill Education.
  • Schroeder, R. G., Clark, M. W., & Cathey, J. M. (2010). Operations Management in the Supply Chain. McGraw-Hill.
  • Simons, R. (2013). Levers of Control: How Managers Use Innovative Control Systems to Drive Strategic Success. Harvard Business Review Press.
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  • Williamson, O. E. (1985). The Economic Institutions of Capitalism. Free Press.

At the end of this analysis, businesses can strategically utilize these mathematical models to optimize operations, improve profitability, and sustain competitive advantages in their respective markets.

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