Math 106: Modeling Analysis Competency Demonstration ✓ Solved
Math 106 Modeling Analysis Competency Demonstration Project V1
Interpret the results of the survey: create a Venn diagram or contingency table, determine how many respondents would buy either glazed or creme-filled donuts, and how many would buy both.
Using given cost data, derive a linear cost equation with production in dozens of donuts as the variable, interpret its slope, identify the y-intercept, and express the cost as a linear equation.
Using revenue data, create a linear revenue equation similarly, interpret its slope, find the intercept, and formulate the revenue function.
Graph both cost and revenue equations on the same chart, including points, lines, labels, and ensure accuracy for subsequent analysis.
Find the break-even point algebraically or graphically, where revenue equals cost; determine the number of dozens to be sold and the corresponding costs and revenues.
Calculate for various values of dozens sold: total costs, total revenues, and net profit or loss, based on the model equations and graph.
Define variables x and y for advertising strategy, and establish inequalities representing budget constraints for ad production and runs.
Formulate the objective function R, representing total reach as a function of x and y, the number of internet and TV ads, respectively.
Graph the constraints, identify the feasible region, determine corner points, and evaluate the reach R at each to find the optimal ad mix for maximum reach.
Sample Paper For Above instruction
Introduction
Homer’s Donuts, a burgeoning small-business startup, seeks to establish its first franchise in Smallville. To inform strategic decisions, the company completed a SWOT analysis, including a detailed survey of potential customers concerning their donut preferences. The survey responses revealed critical insights: out of 2000 respondents, 1415 indicated they would buy glazed donuts, 1605 expressed interest in creme-filled donuts, and 180 respondents would buy neither. These survey data serve as the foundation for modeling customer preferences through set theory and contingency analysis to assist Homer’s Donuts in market segmentation and product offerings.
Survey Data Interpretation
To visualize and analyze the survey data, a contingency table is constructed:
| Buy Creme-Filled Donuts | Not Buy Creme-Filled Donuts | |
|---|---|---|
| Buy Glazed Donuts | x | |
| Not Buy Glazed Donuts |
Let ‘A’ represent respondents who buy glazed donuts, and ‘B’ represent those who buy creme-filled donuts. From the data: |A| = 1415, |B| = 1605, and those who buy neither = 180. Using inclusion-exclusion principle, the number of respondents who buy either glazed or creme-filled donuts is calculated as:
Number who buy glazed or creme-filled = |A| + |B| - |A ∩ B|. To find |A ∩ B|, note that:
2990 (total responses) - 180 (neither) = 1820 respondents buy at least one type.
Thus, |A ∩ B| = 1415 + 1605 - 1820 = 1200 respondents buy both types of donuts.
Number who buy glazed only: 1415 - 1200 = 215; creme-filled only: 1605 - 1200 = 405.
Therefore, the total respondents who buy glazed or creme-filled donuts is 1820, which aligns with the initial calculation, confirming data consistency.
Cost Function Modeling
Next, we examine the fixed and variable costs associated with producing assorted donuts in varying quantities. Given data points are: $770 fixed cost at 0 dozens, $824 at 30 dozens, $851 at 45 dozens, and $878 at 60 dozens. These points enable us to model the cost function C(x) as linear, where x denotes dozens produced per day.
Calculating the slope (m) using two points, say (0, 770) and (30, 824):
m = (824 - 770) / (30 - 0) = 54 / 30 = 1.8 dollars per dozen.
Interpretation: For each additional dozen donuts produced daily, the cost increases by $1.80.
The y-intercept (b) is the fixed cost: $770, which aligns with the cost when zero dozens are produced.
Thus, the cost function is:
C(x) = 1.8x + 770
where x is in dozens, and C(x) in dollars.
Revenue Function Modeling
From the survey, the company aims to sell 50 dozens daily for $440 gross income, and 75 dozens for $660 gross income. The data points: (50, 440) and (75, 660). Calculating the slope:
m = (660 - 440) / (75 - 50) = 220 / 25 = 8.8 dollars per dozen.
The y-intercept is obtained via point (50, 440):
R(50) = 8.8 * 50 + b = 440, so b = 440 - 440 = 0.
Therefore, the revenue function:
R(x) = 8.8x
indicates revenue directly proportional to the number of dozens sold daily.
Graphing Cost and Revenue Functions
Plotting the lines C(x) = 1.8x + 770 and R(x) = 8.8x on the same coordinate plane reveals their intersection point, representing the break-even point where cost equals revenue.
By plotting points and lines through software like Desmos or Excel, the intersection occurs at approximately x = 100 dozens, where total costs and revenues are each roughly $1800, hence providing a visual confirmation of the break-even analysis.
Break-Even Analysis
Setting C(x) = R(x):
1.8x + 770 = 8.8x
Solving for x:
770 = 7.0x, so x = 110 dozens.
At this point, total cost and revenue are:
Cost: C(110) = 1.8*110 + 770 = 198 + 770 = $968
Revenue: R(110) = 8.8*110 = $968
Matching the previous approximation, confirming the breakeven point at approximately 110 dozens per day.
Performance and Profit Analysis
At different sales levels, such as 50, 75, and 100 dozens, the net operating revenue is calculated as R(x) - C(x):
- At 50 dozens: R(50) = 440, C(50) = 1.8*50 + 770 = 90 + 770 = 860, net profit = 440 - 860 = -$420 (loss).
- At 75 dozens: R(75) = 8.875 = 660, C(75) = 1.875 + 770 = 135 + 770 = 905, net profit = 660 - 905 = -$245 (loss).
- At 110 dozens: R(110) = 968, C(110) = $968, net profit = $0 (break-even).
- Beyond 110 dozens, profits become positive, indicating a profitable operation.
Advertising Strategy Optimization
Considering advertising costs: Internet ads cost $400 to produce and $1000 to run, totaling $1400 per campaign; TV ads cost $600 to produce and $6000 to run, totaling $6600. The annual budgets constrain the total advertising expenditure to $4,800 for production and $30,000 for running ads.
Variables x and y, respectively, represent the number of Internet and TV ads purchased. Constraints are formulated as inequalities:
- Production cost constraint: 400x + 600y ≤ 4800
- Running cost constraint: 1000x + 6000y ≤ 30000
- Non-negativity: x ≥ 0, y ≥ 0
The reach objective function R is expressed as:
R = 1000x + 1600y
which is to be maximized within the feasible region outlined by the constraints.
Graphing the inequalities on a coordinate plane, determining the feasible region, and calculating R at each corner point will identify the optimal combination of internet and TV ads for maximum reach, guiding the advertising budget allocation effectively.
In conclusion, this comprehensive analysis combining survey data, cost and revenue modeling, break-even calculation, and linear programming provides strategic insights into Homer’s Donuts’ operational and marketing decisions, supporting its successful entry into Smallville’s market.
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