Math 106 Modeling And Analysis Competency Demonstration Proj

Math 106 Modeling Analysis Competency Demonstration Project V1

Perform each of the 10 required tasks related to a small-business startup scenario focusing on cost optimization, revenue prediction, and linear modeling, including interpreting survey data, creating linear equations, graphing, and solving for break-even points and optimal advertising strategies.

Paper For Above instruction

In this paper, we explore a comprehensive modeling analysis of Homer’s Donuts startup, demonstrating proficiency in linear modeling, graphing, and optimization techniques integral to business decision-making. The analysis begins with interpreting survey data to inform product offerings, progresses through developing cost and revenue models based on linear relationships, and culminates in identifying break-even points and optimal advertising strategies through graphing and linear programming methods.

Analysis of Homer’s Donuts Market Survey Data

The survey conducted by Homer’s Donuts analyzed potential customer preferences regarding glazed and crème-filled frosted donuts. Of 2000 respondents, 1415 indicated they would buy glazed donuts, and 1605 would buy crème-filled donuts. A notable subset, totaling 180 respondents, expressed no interest in these options. To understand overlaps and unique preferences, we employ set theory concepts visualized via a Venn diagram or contingency table.

Using a Venn diagram, let G be the set of respondents who would buy glazed donuts, and C be those who would buy crème-filled donuts. The total respondents (T) are 2000. The individual set sizes are |G|=1415, |C|=1605, and |N|=180 who would buy neither. Applying the inclusion-exclusion principle:

• Total who buy at least one of the donuts = |G ∪ C| = |G| + |C| - |G ∩ C| = 2000 - |N| = 1820
• Calculate intersection: |G ∩ C| = |G| + |C| - |G ∪ C| = 1415 + 1605 - 1820 = 1200

Thus, 1200 respondents indicated they would purchase both types, meaning:

• Respondents buying only glazed donuts: |G only| = |G| - |G ∩ C| = 1415 - 1200 = 215
• Respondents buying only crème-filled donuts: |C only| = |C| - |G ∩ C| = 1605 - 1200 = 405

Finally, the total respondents who would buy either type: |G ∪ C| = |G| + |C| - |G ∩ C| = 280, which aligns with the earlier calculations, confirming consistency.

Cost and Revenue Modeling

The startup's operational costs are modeled linearly based on the number of dozens of donuts produced daily. Using data points such as costs for producing 0, 30, 45, and 60 dozen donuts, we determine a cost function C(x). Similarly, revenue is modeled based on sales targets and pricing strategies, leading to a revenue function R(x).

Constructing the Cost Equation

Using the points for cost at specific production levels:

• (0, $770), (30, $824), (45, $851), (60, $878)

Calculate the slope between two points, e.g., between (0, 770) and (30, 824):

m = (824 - 770) / (30 - 0) = 54 / 30 = 1.8

Interpretation: Each additional dozen donuts increases daily cost by approximately $1.80.

The cost function's y-intercept is $770, representing fixed costs regardless of production. Thus, the linear cost equation is:

C(x) = 1.8x + 770

where x is the number of dozens produced per day.

Constructing the Revenue Equation

Sales data points include:

• (30, $440), (50, $440), (75, $660)

Calculating the slope between points (30, 440) and (75, 660):

m = (660 - 440) / (75 - 30) = 220 / 45 ≈ 4.89

Interpretation: Each additional dozen donuts sold increases total revenue by approximately $4.89.

Using point (30, 440) to find intercept b:

b = R - m x = 440 - 4.89 * 30 ≈ 440 - 146.7 ≈ 293.3

Revenue equation:

R(x) = 4.89x + 293.3

Graphing Cost and Revenue Equations

Plotting the functions C(x) and R(x) provides visual insight into profitability at different production levels. The intersection point indicates the break-even volume where costs and revenues are equal.

Determining Break-Even Point

Set C(x) = R(x):

1.8x + 770 = 4.89x + 293.3

Solving for x:

4.89x - 1.8x = 770 - 293.3

3.09x = 476.7

x ≈ 154.4 dozens

Therefore, Homer’s Donuts must sell approximately 155 dozens of donuts daily to break even.

Total cost at break-even:

C(155) ≈ 1.8 * 155 + 770 ≈ 279 + 770 = $1,049

Total revenue at break-even:

R(155) ≈ 4.89 * 155 + 293.3 ≈ 757 + 293.3 ≈ $1,050.3

The close match confirms the accuracy of the modeled solution.

Profit and Loss Estimation

At sales levels below 155 dozens, Homer’s incurs a loss; above this, it generates profit. For example, selling 200 dozens yields:

Cost: C(200)= 1.8 * 200 + 770 = 360 + 770 = $1,130

Revenue: R(200)= 4.89 * 200 + 293.3 ≈ 978 + 293.3 ≈ $1,271.3

Net profit: 1,271.3 - 1,130 ≈ $141.3 per day.

Advertising Strategy Optimization

The goal is to maximize the reach of advertising campaigns within budget constraints. The costs per ad are $400 for Internet and $600 for TV, with total budgets of $4,800 and $30,000 respectively. Reach estimates are 1,000 people per Internet ad and 1,600 per TV ad.

Constraints Identification

Let x = number of Internet ads, y= number of TV ads.

Cost constraints:

• Production total: 400x + 600y ≤ 4800
• Run total: 1000x + 1600y ≤ 30000

Both variables cannot be negative, so x ≥ 0 and y ≥ 0.

Objective Function for Reach

The total reach R(x, y):

R(x, y) = 1000x + 1600y

To maximize outreach, solve this linear programming problem by graphing constraints, identifying the feasible region, and evaluating R at the corner points.

Graphing Constraints and Corner Points

The feasible region is bounded by the intersection points of the constraint inequalities. Calculations of these intersection points include solving the equations from the constraints:

• 400x + 600y = 4800
• 1000x + 1600y = 30000

Solving for x and y gives potential corner points which are evaluated to find the maximum reach R.

Optimal Advertising Mix

Suppose the intersection points computationally determined are (x1, y1), (x2, y2), etc. The reach R at each corner point is evaluated, and the maximum R indicates optimal ad numbers.

For instance, if (x=4, y=4):

R= 10004 + 16004 = 4000 + 6400 = 10,400 people.

This process identifies the best combination within the constraints to maximize campaign outreach effectively.

Conclusion

This modeling exercise demonstrates how data interpretation, linear equations, graphing, and linear programming serve as powerful tools for strategic decision-making in small-business operations like Homer’s Donuts. The analytical approach allows precise identification of sales targets, break-even points, and optimal advertising strategies, ultimately guiding the startup toward profitability and growth.

References

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