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The “Modeling & Analysis Competency Demonstration Project” is designed to assess your ability to use the skills in linear modeling, linear systems & programming, and counting techniques that you’ve learned in a practical real-world application. The project scenario is a small-business startup and does not focus on EVERY aspect of a small business startup, but does focus on cost optimization and profit/loss calculation and prediction through the use of linear modeling and linear programming.
Instructions:
If you have any questions about this project, please post them in the “Modeling & Analysis Competency Demonstration Project Discussion” topic found at Discussions > Modeling & Analysis Competency Demonstration Project forum in our LEO classroom so everyone can see what we’re talking about. Good luck on the project!
Business Case Study – Homer’s Donuts
Homer’s Donuts is a small-business startup looking to open its first franchise location in Smallville. Your ownership team completed its business and marketing SWOT (“strengths – weaknesses – opportunities – threats”) analyses and believes there’s a real chance for Homer’s Donuts to be a successful business!
The SWOT marketing analysis included a survey of potential customers in and around the Smallville metropolitan area. The survey asked people what pastries they would most likely buy when they stop for coffee on the way to wherever they were going. 2000 people responded to the survey, and the results are as follows:
______________________________________________________________________
Task 1: Interpreting the Marketing Survey: Counting
For the survey results above:
****************************************************************************************************
From your SWOT analyses, your team decides the best way to go is to offer assorted donuts, sold individually or by the dozen. Now your team has to come up with a daily operation cost/revenue analysis to see if it’s actually worth going into business. Here’s what the team knows so far:
______________________________________________________________________
Task 2: Determine Cost Equation
Use the information in paragraph A above to:
______________________________________________________________________
Task 3: Determine Revenue Equation
Use the information in paragraph B above to:
______________________________________________________________________
Task 4: Graph Cost and Revenue Equations on Same Chart
Using the information you created in Tasks 2 and 3, create a single chart that shows:
Use an appropriate scale on the horizontal and vertical axes of your chart, make sure the correct values go with the correct axis, and be sure to label carefully (chart title, axes titles, etc). You may create your chart using software (MS Excel, DESMOS, etc) or you may draw it by hand. This chart will help you answer task 5 requirements, so do your best to make it accurate!
If you use Microsoft Excel, be sure to check out Content > Course Resources > Webliography in our LEO classroom for two topics on scatterplot construction and linear regression which are very helpful when creating a chart. If you want to use the free DESMOS graphing software to create your chart, be sure to check out Content > Course Resources > Webliography > “Desmos” Free Online Graphing Calculator in our LEO classroom to access DESMOS, make your chart on that site, and save the image to upload into your project submission. Be sure it’s labeled correctly!If you draw by hand, the chart MUST be proportionally accurate (that means use graph paper as a minimum), all points and graphed lines must be identified, and axes and scales must be correctly labeled.
When completed, your chart depicts the “cost/revenue model” for Homer’s Donuts!
****************************************************************************************************
Your team knows that Homer’s Donuts has to sell a minimum number of dozens of assorted donuts every day before it starts to make a profit. That minimum number of dozens of assorted donuts needing to be sold every day is defined as the break-even point.
Task 5: Break-Even Point Determination
Identify the number of dozens of assorted donuts x that need to be sold per day in order for daily revenue to equal daily production costs [which means that value of x which makes R(x) = C(x) ]. Also identify the total daily cost and total daily revenue at the break-even point. You may determine the break-even point graphically (using chart you created in Task 4) or algebraically by making a system of the two linear equations you created in Tasks 2 and 3 and solving that system. Answers must be exact !
Break-even occurs when x = _____________ dozens of assorted donuts are sold daily.
At break-even point, C(x) = $ ______________ per day
At break-even point, R(x) = $ ______________ per day
______________________________________________________________________
Task 6: Using Model to Determine Performance and Net Operating Revenue (Profit or Loss)
Using the “cost/revenue model” graph created in Task 4 and/or the equations created in Tasks 2 and 3, determine how Homer’s Donuts would perform by completing the following table (fill in all the blanks – give answers rounded to the nearest dollar):
|
If # dozen assorted donuts sold daily x = |
Then total daily production cost C(x) (in $) = |
And daily total (gross) revenue R(x) (in $) = |
And net daily operating revenue R(x) – C(x) (in $) = |
|
80 |
|
|
|
|
110 |
|
|
|
|
200 |
|
|
|
|
500 |
|
|
|
For net daily operating revenue: Positive values represent PROFIT; negative values represent LOSS
****************************************************************************************************
Now it’s time to look at the advertising budget for Homer’s Donuts!
The advertising agency you hired presented the following costs for Internet and television ads for Homer’s Donuts. Producing an Internet ad costs $400 and producing a television ad costs $600. The cost to run an Internet ad is $1000, and it will cost $6000 to run a TV ad. Your team decides that the total amount to be spent on ads is not to exceed $4,800 for production and not to exceed $30,000 for running the ads. The agency estimates that each viewing of an Internet ad will reach 1000 people and each airing of a television ad will reach 1600 people. How many of each type of ad should be bought so that the total number of people reached is a maximum?
Linear programming is the mathematical method best used to determine the values of input variables needed to optimize the “objective function”, whether that function is maximizing the number of people an ad campaign reaches or minimizing the costs associated with the operation of a manufacturing process.
Task 7: Identify Constraints on the Process
From the scenario, identify what variables x and y will represent.Using the information above fill in the following table, and then use the table to create the two algebraic equations or inequalities that identify the constraints on the process of advertising for Homer’s Donuts:All constraint linear equations/inequalities graphed“Feasible region” with its boundariesCoordinates of each “corner point” of the feasible region
Use an appropriate scale on the horizontal and vertical axes of your chart, make sure the correct values go with the correct axis, and be sure to label carefully (chart title, axes titles, etc). You may create your chart using graphing calculators (DESMOS, TI-80 series graphing calculators, etc) or you may draw it by hand. Do your best to make it accurate!
If you want to use the free DESMOS graphing software to create your chart, be sure to check out Content > Course Resources > Webliography > “Desmos” Free Online Graphing Calculator in our LEO classroom to access DESMOS, make your chart on that site, and save the image to upload into your project submission. Be sure it’s labeled correctly!If you draw by hand, the chart MUST be proportionally accurate (that means use graph paper as a minimum), all points and graphed lines must be identified, and axes and scales must be correctly labeled.
______________________________________________________________________
Task 10: Optimizing the Objective Function
Using the objective function determined in Task 8 and the graph created in Task 9:
Create a corner point table by filling in the blanks of the following table (enter as many rows as you need). For each corner point entered in the left column, evaluate the objective function by entering the corresponding corner point values for x and y and solve for R. Enter value of R in right columnDetermine which corner point coordinate pair optimizes the objective function:If you want to maximize R, select the corner point pair that produces the greatest value of R If you want to minimize R, select the corner point pair that produces the smallest value of R.
The corner point pair you select represents the number of Internet ads to be created and the number of TV adds to be created to result in maximum “reach” R. You cannot “mix” corner point values; you must use the x and y values of the corner point you select.
Number of Internet ads that should be bought to maximize “reach” R : ____________
Number of times TV ads that should be bought to maximize “reach” R : ___________
****************************************************************************************************The “Modeling & Analysis Competency Demonstration Project” isdesigned to assess your ability to use the skills in linear modeling, linearsystems & programming, and counting techniques that you’ve learned in apractical real-world application. The projectscenario is a small-business startup and does not focus on EVERY aspect of asmall business startup, but does focus on cost optimization and profit/losscalculation and prediction through the use of linear modeling and linearprogramming.
Instructions:
1. Access (and download if you like) the scenario (in either MSWord of PDF format) from Content >Competency Demonstration Projects > Modeling & Analysis CompetencyDemonstration Project in our LEO classroom.
2. Perform each of the 10 required tasks to the best of yourability. Note that some of the tasks require calculations, others requirelinear equation construction, still others require graphing. PLEASE show all your work when doingcalculations, equation construction and graphing if you want me to considerawarding partial credit! I cannot givepartial credit if you don’t show your work!
3. Compile your submission. Be sure your submission includes:
a. Answers to the questions posed in each of the 10 tasks;
b. Required graphs (2) and whatever supporting graphs you wishto include
c. Whatever work supporting your answers that you wish me toconsider for partial credit (of course, if you don’t submit any work and youranswer is incorrect, then I’m obliged to award no credit for your answer…)
4. Upload your submission into the “Modeling & AnalysisCompetency Demonstration Project” folder in the Assignments link under the NavBar in our LEO classroom. Besure to forward your initial submission no later than 11:59PM Eastern time onthe Sunday ending the 5th week of class if you want to beeligible for full credit for this project! Any initial submissions received after this deadline are only eligiblefor a maximum of 75 out of 100 points (see “Grading Criteria” section of Content > Course Syllabus for moredetails).
5. You’ll receive an initial grade and feedback (grade will bevisible in your Gradebook; grade and feedback will be visible in yourAssignment Folder). You are free (andencouraged!) to make changes to your submission based on the feedback you receiveand then resubmit your project for a better score. You may do this as many times as you wishprior to 11:59PM Eastern time on the Thursday of Week 8. I will grade resubmissions and post yourupdated project grade as well as updated feedback (if necessary).
If you have any questions about this project, please post them in the“Modeling & Analysis Competency Demonstration Project Discussion” topicfound at Discussions > Modeling &Analysis Competency Demonstration Project forum in our LEO classroom so everyonecan see what we’re talking about. Goodluck on the project!
Business Case Study– Homer’s Donuts
Homer’s Donuts is asmall-business startup looking to open its first franchise location inSmallville. Your ownership teamcompleted its business and marketing SWOT (“strengths – weaknesses –opportunities – threats”) analyses and believes there’s a real chance forHomer’s Donuts to be a successful business!
The SWOT marketinganalysis included a survey of potential customers in and around the Smallvillemetropolitan area. The survey asked peoplewhat pastries they would most likely buy when they stop for coffee on the wayto wherever they were going. 2000 peopleresponded to the survey, and the results are as follows:
· 1415 respondents saidthey would buy glazed donuts
· 1605 respondentssaid they would buy creme-filled frosted donuts
· 180 respondentssaid they would buy neither glazed nor creme-filled frosted donuts
______________________________________________________________________
Task 1: Interpreting the Marketing Survey: Counting
For the survey results above:
a) Create either a Venn diagram or a contingency tabledepicting the results of the survey.
b) Determine how many of the survey respondents would buyeither a glazed donut or a creme-filled frosted donut?
c) Determine how many of the survey respondents would buy aglazed donut and a creme-filled frosted donut?
****************************************************************************************************
From your SWOT analyses,your team decides the best way to go is to offer assorted donuts, soldindividually or by the dozen. Now yourteam has to come up with a daily operation cost/revenue analysis to see if it’sactually worth going into business. Here’s what the team knows so far:
A. Before making any donuts, there is a $770 fixed cost eachday (which covers the building and its equipment, as well as labor costs for 2shifts of 4 employees each). That meansthe cost to produce 0 dozen assorted donuts per day is $770. The cost to produce 30 dozen assorted donutsper day is $824; the cost to produce 45 dozen assorted donuts per day is $851;and the cost to produce 60 dozen assorted donuts per day is $878. The relationship between dozens of donutsproduced per day and cost per day is linear.
B. To undercut the competition, your team recommends settingthe price per dozen assorted donuts so that Homer’s gets $440 total (gross)income for 50 dozen assorted donuts sold per day, and $660 total (gross) incomefor 75 dozen assorted donuts sold per day. The relationship between dozen donuts sold perday and total (gross) income per day is linear.
______________________________________________________________________
Task 2: DetermineCost Equation
Use the information in paragraph A above to:
· Create at least twoordered pairs. Let the x value be the number of dozen donutsproduced per day and the y value bethe cost to produce them.
· Use two of the orderedpairs to determine the slope m of thelinear equation that describes daily operating cost C(x) where x representsnumber of dozens of donuts produced per day. Interpret the meaning of the slope.
· Determine the y – intercept value (also known as b)
· Use what you’vefound to create the linear equation in y= mx + b (slope-intercept form) that describes daily production costs forHomer’s Donuts
· Take the equation youcreated and replace “y” with “C(x)” to create the cost equation.
______________________________________________________________________
Task 3: DetermineRevenue Equation
Use the information in paragraph B above to:
· Create at least twoordered pairs. Let the x value be the number of dozen donutsproduced per day and the y value bethe total income (revenue) from selling them.
· Use two of theordered pairs to determine the slope mof the linear equation that describes daily revenue R(x) where x representsnumber of dozens of donuts produced per day. Interpret the meaning of the slope.
· Determine the y – intercept value (also known as b)
· Use what you’vefound to create the linear equation in y= mx + b (slope-intercept form) that describes daily revenue for Homer’sDonuts
· Take the equationyou created and replace “y” with “R(x)” to create the revenue equation.
______________________________________________________________________
Task 4: Graph Cost and Revenue Equations on SameChart
Using the information you created in Tasks 2 and 3, create a singlechart that shows:
a) Ordered pair points for daily production cost you created inTask 2
b) Ordered pair points for daily revenue you created in Task 3
c) Graph of the cost equation you created in Task 2
d) Graph of the revenue equation you created in Task 3
Use an appropriate scale on the horizontal and vertical axes of yourchart, make sure the correct values go with the correct axis, and be sure tolabel carefully (chart title, axes titles, etc). You may create your chart using software (MSExcel, DESMOS, etc) or you may draw it by hand. This chart will help you answer task 5 requirements, so do your best tomake it accurate!
o If you use Microsoft Excel, be sure to check out Content > Course Resources >Webliography in our LEO classroom for two topics on scatterplotconstruction and linear regression which are very helpful when creating achart.
o If you want to use the free DESMOS graphing software tocreate your chart, be sure to check out Content> Course Resources > Webliography > “Desmos” Free Online GraphingCalculator in our LEO classroom to access DESMOS, make your chart on thatsite, and save the image to upload into your project submission. Be sure it’s labeled correctly!
o If you draw by hand, the chart MUST be proportionallyaccurate (that means use graph paper as a minimum), all points and graphedlines must be identified, and axes and scales must be correctly labeled.
When completed, your chart depicts the “cost/revenue model” for Homer’sDonuts!
****************************************************************************************************
Your team knows thatHomer’s Donuts has to sell a minimum number of dozens of assorted donuts everyday before it starts to make a profit. That minimum number of dozens of assorted donuts needing to be soldevery day is defined as the break-evenpoint.
Task 5: Break-Even Point Determination
Identify the number ofdozens of assorted donuts x that needto be sold per day in order for daily revenue to equal daily production costs[which means that value of x whichmakes R(x) = C(x) ]. Also identify the total daily cost and totaldaily revenue at the break-even point. Youmay determine the break-even point graphically (using chart you created in Task4) or algebraically by making a system of the two linear equations you createdin Tasks 2 and 3 and solving that system. Answers must be exact !
Break-even occurs when x =_____________ dozens of assorted donuts are sold daily.
At break-even point, C(x) = $ ______________ per day
At break-even point, R(x) = $ ______________ per day
______________________________________________________________________
Task 6: Using Model to Determine Performance and Net OperatingRevenue (Profit or Loss)
Using the “cost/revenue model” graph created in Task 4 and/or theequations created in Tasks 2 and 3, determine how Homer’s Donuts would performby completing the following table (fillin all the blanks – give answers rounded to the nearest dollar):
|
If # dozen assorted donuts sold daily x =
|
Then total daily production cost C(x) (in $) =
|
And daily total (gross) revenue R(x) (in $) =
|
And net daily operating revenue R(x) – C(x) (in $) =
|
|
80
|
|
|
|
|
110
|
|
|
|
|
200
|
|
|
|
|
500
|
|
|
|
For net daily operating revenue: Positive values represent PROFIT; negative values represent LOSS
****************************************************************************************************
Now it’s time to look at the advertising budget for Homer’s Donuts!
The advertising agency youhired presented the following costs for Internet and television ads for Homer’sDonuts. Producing an Internet ad costs $400and producing a television ad costs $600. The cost to run an Internet ad is $1000, and it will cost $6000 to run aTV ad. Your team decides that the totalamount to be spent on ads is not to exceed $4,800 for production and not toexceed $30,000 for running the ads. Theagency estimates that each viewing of an Internet ad will reach 1000 people andeach airing of a television ad will reach 1600 people. How many of each type of ad should be boughtso that the total number of people reached is a maximum?
Linear programming isthe mathematical method best used to determine the values of input variables neededto optimize the “objective function”, whether that function is maximizing thenumber of people an ad campaign reaches or minimizing the costs associated withthe operation of a manufacturing process.
Task 7: Identify Constraints on the Process
· From the scenario,identify what variables x and y will represent.
· Using theinformation above fill in the following table, and then use the table to createthe two algebraic equations or inequalities that identify the constraints onthe process of advertising for Homer’s Donuts:
|
Ad Process Task
|
Cost per Internet ad ($)
|
# of Internet ads
|
Cost per TV ad ($)
|
# of TV ads
|
Enter correct sign (
|
Total Cost ($)
|
|
Production
|
|
|
|
|
|
|
|
Running
|
|
|
|
|
|
|
Constraint on producing ads: _______________________________________
Constraint on running of ads: ________________________________________
If the variables x and y cannot be negative, be sure to include and as two additional constraints!
______________________________________________________________________
Task 8: Identify Objective Function
Algebraically define the objectivefunction, which is the “reach” R thatthis ad campaign will have. Fill in thefollowing table, and then use the table to create the algebraic “objectivefunction” that defines the reach R (interms of x and y) for Homer’s Donuts advertising campaign:
|
# people reached by Internet ad
|
# of Internet ads
|
Enter correct sign (+, – )
|
# people reached by TV ad
|
# of TV ads
|
Enter correct sign (
|
Reach (R)
|
|
|
|
|
|
|
|
R
|
Algebraic form of objective function: R = _______________________________
______________________________________________________________________
Task 9: Graph Constraints, Identify Feasible Regionand Corner Points
Using the constraints you defined in Task 7, construct a graph with thefollowing features identified:
· All constraintlinear equations/inequalities graphed
· “Feasible region”with its boundaries
· Coordinates of each“corner point” of the feasible region
Use an appropriate scale on the horizontal and vertical axes of yourchart, make sure the correct values go with the correct axis, and be sure tolabel carefully (chart title, axes titles, etc). You may create your chart using graphingcalculators (DESMOS, TI-80 series graphing calculators, etc) or you may draw itby hand. Do your best to make itaccurate!
o If you want to use the free DESMOS graphing software tocreate your chart, be sure to check out Content> Course Resources > Webliography > “Desmos” Free Online GraphingCalculator in our LEO classroom to access DESMOS, make your chart on thatsite, and save the image to upload into your project submission. Be sure it’s labeled correctly!
o If you draw by hand, the chart MUST be proportionallyaccurate (that means use graph paper as a minimum), all points and graphedlines must be identified, and axes and scales must be correctly labeled.
______________________________________________________________________
Task 10: Optimizing the Objective Function
Using the objective function determined in Task 8 and the graph createdin Task 9:
· Create a cornerpoint table by filling in the blanks of the following table (enter as many rowsas you need). For each corner pointentered in the left column, evaluate the objective function by entering thecorresponding corner point values for xand y and solve for R. Enter value of R in rightcolumn
|
Corner Point Coordinates (x , y)
|
Enter objective function:
R =
|
|
|
|
|
|
|
|
|
|
|
|
|
· Determine whichcorner point coordinate pair optimizes the objective function:
o If you want to maximize R,select the corner point pair that producesthe greatest value of R
o If you want to minimize R,select the corner point pair that produces the smallest value of R.
The corner point pair you select represents the number of Internet adsto be created and the number of TV adds to be created to result in maximum“reach” R. You cannot “mix” corner point values; youmust use the x and y values of the corner point you select.
Number of Internet ads that should be bought to maximize “reach” R : ____________
Number of times TV ads that should be bought to maximize “reach” R : ___________
****************************************************************************************************
MATH 106 Modeling & Analysis Competency Demonstration Project V1.0 (5/16) The “Modeling & Analysis Competency Demonstration Project” is designed to assess
your ability to use the skills in linear modeling, linear systems & programming, and
counting techniques that you’ve learned in a practical real-world application. The
project scenario is a small-business startup and does not focus on EVERY aspect of a
small business startup, but does focus on cost optimization and profit/loss calculation
and prediction through the use of linear modeling and linear programming.
Instructions:
1. Access (and download if you like) the scenario (in either MS Word of PDF format)
from Content > Competency Demonstration Projects > Modeling & Analysis
Competency Demonstration Project in our LEO classroom.
2. Perform each of the 10 required tasks to the best of your ability. Note that some
of the tasks require calculations, others require linear equation construction, still
others require graphing. PLEASE show all your work when doing calculations,
equation construction and graphing if you want me to consider awarding partial
credit! I cannot give partial credit if you don’t show your work!
3. Compile your submission. Be sure your submission includes:
a. Answers to the questions posed in each of the 10 tasks;
b. Required graphs (2) and whatever supporting graphs you wish to include
c. Whatever work supporting your answers that you wish me to consider for
partial credit (of course, if you don’t submit any work and your answer is
incorrect, then I’m obliged to award no credit for your answer…)
4. Upload your submission into the “Modeling & Analysis Competency
Demonstration Project” folder in the Assignments link under the NavBar in our
LEO classroom. Be sure to forward your initial submission no later than
11:59PM Eastern time on the Sunday ending the 5th week of class if you
want to be eligible for full credit for this project! Any initial submissions received
after this deadline are only eligible for a maximum of 75 out of 100 points (see
“Grading Criteria” section of Content > Course Syllabus for more details).
5. You’ll receive an initial grade and feedback (grade will be visible in your
Gradebook; grade and feedback will be visible in your Assignment Folder). You
are free (and encouraged!) to make changes to your submission based on the
feedback you receive and then resubmit your project for a better score. You may
do this as many times as you wish prior to 11:59PM Eastern time on the
Thursday of Week 8. I will grade resubmissions and post your updated project
grade as well as updated feedback (if necessary).
If you have any questions about this project, please post them in the “Modeling &
Analysis Competency Demonstration Project Discussion” topic found at Discussions >
Modeling & Analysis Competency Demonstration Project forum in our LEO
classroom so everyone can see what we’re talking about. Good luck on the project! 1 MATH 106 Modeling & Analysis Competency Demonstration Project V1.0 (5/16) Business Case Study – Homer’s Donuts
Homer’s Donuts is a small-business startup looking to open its first franchise
location in Smallville. Your ownership team completed its business and marketing
SWOT (“strengths – weaknesses – opportunities – threats”) analyses and believes
there’s a real chance for Homer’s Donuts to be a successful business!
The SWOT marketing analysis included a survey of potential customers in and
around the Smallville metropolitan area. The survey asked people what pastries they
would most likely buy when they stop for coffee on the way to wherever they were
going. 2000 people responded to the survey, and the results are as follows: 1415 respondents said they would buy glazed donuts
1605 respondents said they would buy creme-filled frosted donuts
180 respondents said they would buy neither glazed nor creme-filled frosted
donuts ______________________________________________________________________
Task 1: Interpreting the Marketing Survey: Counting
For the survey results above:
a) Create either a Venn diagram or a contingency table depicting the results of the
survey.
b) Determine how many of the survey respondents would buy either a glazed donut
or a creme-filled frosted donut?
c) Determine how many of the survey respondents would buy a glazed donut and a
creme-filled frosted donut? ****************************************************************************************************
From your SWOT analyses, your team decides the best way to go is to offer
assorted donuts, sold individually or by the dozen. Now your team has to come up with
a daily operation cost/revenue analysis to see if it’s actually worth going into business.
Here’s what the team knows so far: A. Before making any donuts, there is a $770 fixed cost each day (which covers the
building and its equipment, as well as labor costs for 2 shifts of 4 employees
each). That means the cost to produce 0 dozen assorted donuts per day is
$770. The cost to produce 30 dozen assorted donuts per day is $824; the cost to
produce 45 dozen assorted donuts per day is $851; and the cost to produce 60 2 MATH 106 Modeling & Analysis Competency Demonstration Project V1.0 (5/16) dozen assorted donuts per day is $878. The relationship between dozens of
donuts produced per day and cost per day is linear.
B. To undercut the competition, your team recommends setting the price per dozen
assorted donuts so that Homer’s gets $440 total (gross) income for 50 dozen
assorted donuts sold per day, and $660 total (gross) income for 75 dozen
assorted donuts sold per day. The relationship between dozen donuts sold per
day and total (gross) income per day is linear.
______________________________________________________________________
Task 2: Determine Cost Equation
Use the information in paragraph A above to: Create at least two ordered pairs. Let the x value be the number of dozen donuts
produced per day and the y value be the cost to produce them.
Use two of the ordered pairs to determine the slope m of the linear equation that
describes daily operating cost C(x) where x represents number of dozens of
donuts produced per day. Interpret the meaning of the slope.
Determine the y – intercept value (also known as b)
Use what you’ve found to create the linear equation in y = mx + b (slope-intercept
form) that describes daily production costs for Homer’s Donuts
Take the equation you created and replace “y” with “C(x)” to create the cost
equation. ______________________________________________________________________ Task 3: Determine Revenue Equation
Use the information in paragraph B above to: Create at least two ordered pairs. Let the x value be the number of dozen donuts
produced per day and the y value be the total income (revenue) from selling
them.
Use two of the ordered pairs to determine the slope m of the linear equation that
describes daily revenue R(x) where x represents number of dozens of donuts
produced per day. Interpret the meaning of the slope.
Determine the y – intercept value (also known as b)
Use what you’ve found to create the linear equation in y = mx + b (slope-intercept
form) that describes daily revenue for Homer’s Donuts
Take the equation you created and replace “y” with “R(x)” to create the revenue
equation. ______________________________________________________________________ 3 MATH 106 Modeling & Analysis Competency Demonstration Project V1.0 (5/16) Task 4: Graph Cost and Revenue Equations on Same Chart
Using the information you created in Tasks 2 and 3, create a single chart that shows:
a)
b)
c)
d) Ordered pair points for daily production cost you created in Task 2
Ordered pair points for daily revenue you created in Task 3
Graph of the cost equation you created in Task 2
Graph of the revenue equation you created in Task 3 Use an appropriate scale on the horizontal and vertical axes of your chart, make sure
the correct values go with the correct axis, and be sure to label carefully (chart title,
axes titles, etc). You may create your chart using software (MS Excel, DESMOS, etc) or
you may draw it by hand. This chart will help you answer task 5 requirements, so do
your best to make it accurate!
o If you use Microsoft Excel, be sure to check out Content > Course
Resources > Webliography in our LEO classroom for two topics on
scatterplot construction and linear regression which are very helpful when
creating a chart.
o If you want to use the free DESMOS graphing software to create your chart,
be sure to check out Content > Course Resources > Webliography >
“Desmos” Free Online Graphing Calculator in our LEO classroom to
access DESMOS, make your chart on that site, and save the image to upload
into your project submission. Be sure it’s labeled correctly!
o If you draw by hand, the chart MUST be proportionally accurate (that means
use graph paper as a minimum), all points and graphed lines must be
identified, and axes and scales must be correctly labeled. When completed, your chart depicts the “cost/revenue model” for Homer’s Donuts! **************************************************************************************************** Your team knows that Homer’s Donuts has to sell a minimum number of dozens
of assorted donuts every day before it starts to make a profit. That minimum number of
dozens of assorted donuts needing to be sold every day is defined as the break-even
point. 4 MATH 106 Modeling & Analysis Competency Demonstration Project V1.0 (5/16) Task 5: Break-Even Point Determination
Identify the number of dozens of assorted donuts x that need to be sold per day
in order for daily revenue to equal daily production costs [which means that value of x
which makes R(x) = C(x) ]. Also identify the total daily cost and total daily revenue at
the break-even point. You may determine the break-even point graphically (using chart
you created in Task 4) or algebraically by making a system of the two linear equations
you created in Tasks 2 and 3 and solving that system. Answers must be exact ! Break-even occurs when x = _____________ dozens of assorted donuts are sold daily. At break-even point, C(x) = $ ______________ per day At break-even point, R(x) = $ ______________ per day ______________________________________________________________________ Task 6: Using Model to Determine Performance and Net Operating Revenue
(Profit or Loss)
Using the “cost/revenue model” graph created in Task 4 and/or the equations created in
Tasks 2 and 3, determine how Homer’s Donuts would perform by completing the
following table (fill in all the blanks – give answers rounded to the nearest dollar): If # dozen
assorted donuts
sold daily x =
80
110
200
500 Then total daily
production cost
C(x) (in $) = And daily total
(gross) revenue
R(x) (in $) = And net daily
operating revenue
R(x) – C(x) (in $) = For net daily operating revenue: Positive values represent PROFIT; negative values
represent LOSS 5 MATH 106 Modeling & Analysis Competency Demonstration Project V1.0 (5/16) ****************************************************************************************************
Now it’s time to look at the advertising budget for Homer’s Donuts!
The advertising agency you hired presented the following costs for Internet and
television ads for Homer’s Donuts. Producing an Internet ad costs $400 and producing
a television ad costs $600. The cost to run an Internet ad is $1000, and it will cost
$6000 to run a TV ad. Your team decides that the total amount to be spent on ads is
not to exceed $4,800 for production and not to exceed $30,000 for running the ads.
The agency estimates that each viewing of an Internet ad will reach 1000 people and
each airing of a television ad will reach 1600 people. How many of each type of ad
should be bought so that the total number of people reached is a maximum?
Linear programming is the mathematical method best used to determine the
values of input variables needed to optimize the “objective function”, whether that
function is maximizing the number of people an ad campaign reaches or minimizing the
costs associated with the operation of a manufacturing process. Task 7: Identify Constraints on the Process From the scenario, identify what variables x and y will represent. Using the information above fill in the following table, and then use the
table to create the two algebraic equations or inequalities that identify the
constraints on the process of advertising for Homer’s Donuts: Ad Process
Task Cost per
Internet ad
($) # of
Internet
ads Cost per
TV ad ($) # of
TV ads Enter correct
sign
(
≤ ,<,=,> ,≥ ¿ Total
Cost ($) Production
Running
Constraint on producing ads: _______________________________________
Constraint on running of ads: ________________________________________ If the variables x and y cannot be negative, be sure to include
two additional constraints! 6 x≥0 and y≥0 as MATH 106 Modeling & Analysis Competency Demonstration Project V1.0 (5/16) ______________________________________________________________________
Task 8: Identify Objective Function
Algebraically define the objective function, which is the “reach” R that this ad
campaign will have. Fill in the following table, and then use the table to create the
algebraic “objective function” that defines the reach R (in terms of x and y) for Homer’s
Donuts advertising campaign:
# people
reached by
Internet ad # of
Internet
ads Enter
correct
sign (+, – ) # people
reached
by TV ad # of
TV
ads Enter correct
sign
(
≤ ,<,=,> ,≥ ¿ Reac
h (R) R Algebraic form of objective function: R = _______________________________ ______________________________________________________________________
Task 9: Graph Constraints, Identify Feasible Region and Corner Points
Using the constraints you defined in Task 7, construct a graph with the following
features identified: All constraint linear equations/inequalities graphed
“Feasible region” with its boundaries
Coordinates of each “corner point” of the feasible region Use an appropriate scale on the horizontal and vertical axes of your chart, make sure
the correct values go with the correct axis, and be sure to label carefully (chart title,
axes titles, etc). You may create your chart using graphing calculators (DESMOS, TI-80
series graphing calculators, etc) or you may draw it by hand. Do your best to make it
accurate!
o If you want to use the free DESMOS graphing software to create your chart,
be sure to check out Content > Course Resources > Webliography >
“Desmos” Free Online Graphing Calculator in our LEO classroom to
access DESMOS, make your chart on that site, and save the image to upload
into your project submission. Be sure it’s labeled correctly!
o If you draw by hand, the chart MUST be proportionally accurate (that means
use graph paper as a minimum), all points and graphed lines must be
identified, and axes and scales must be correctly labeled. 7 MATH 106 Modeling & Analysis Competency Demonstration Project V1.0 (5/16) ______________________________________________________________________
Task 10: Optimizing the Objective Function
Using the objective function determined in Task 8 and the graph created in Task 9: Create a corner point table by filling in the blanks of the following table (enter as
many rows as you need). For each corner point entered in the left column,
evaluate the objective function by entering the corresponding corner point values
for x and y and solve for R. Enter value of R in right column
Corner Point Coordinates (x , y) Enter objective function:
R= Determine which corner point coordinate pair optimizes the objective function:
o If you want to maximize R, select the corner point pair that produces the
greatest value of R
o If you want to minimize R, select the corner point pair that produces the
smallest value of R. The corner point pair you select represents the number of Internet ads to be created
and the number of TV adds to be created to result in maximum “reach” R. You cannot
“mix” corner point values; you must use the x and y values of the corner point you
select. Number of Internet ads that should be bought to maximize “reach” R : ____________ Number of times TV ads that should be bought to maximize “reach” R : ___________ **************************************************************************************************** 8