Name:___________________
Come to
boulangerie
La Miette
Specializing in Croissants
Plain or chocolate
240 9th ave.
LaSalle, Quebec
Open 7 days a week from 7am to 6 pm
How many of each type?
Pierre and Mary Miette own a small bakery that makes the best croissants. They pride
themselves on only making two types, perfectly. They make plain croissants and chocolate
covered croissants. Their challenge is to figure out how many dozens of each type they need to
produce each morning.
What they know:
The Miettes know that each dozen of their plain croissants requires .75 kg of dough, and that
each dozen of chocolate croissant requires .5 kg. of dough as well as 200 grams of chocolate
(.2kg). They also know that each dozen plain requires 12 minutes (.2 hrs.) to prepare and each
chocolate takes 3 minutes more (15 minutes or .25 hrs.)
Their ability to produce is limited by four factors.
1. Their supplies – they have 200 kg of dough and 45 kg of chocolate at most.
2. Oven space and rack space - they have oven and cooling space for 300 dozen.
3. Labour – they have a maximum of 75 man hours of prep time available.
4. Demand – they know that the most plain croissants they can sell is 260 dozen.
The Miettes wish to produce a number of each type of croissant that will allow them to make
the most money possible taking costs into consideration. The plain croissants sell for $18 a
dozen and cost $8 to make. The chocolate croissants sell for $24 a dozen and cost $11 to make.
How many dozens of each type of croissant should they produce in order to make as much
profit as possible?
Your Assignment:
You own a consulting firm for small businesses, and the Miettes have hired you to help them
optimize their production based on the information we have learned. Not only do you want to
give them the best advice, you also have to effectively communicate the information to them in
clear mathematical terms, and appropriate visual aids. Use your newly acquired math skills to
prepare a presentation for the Miettes. You should include:
1. A solution to the Miette’s question including the equations involving quantities of
ingredients, oven space, and prep time, including the amount of croissants they will be
baking.
2. A graphic representation of the situation.
3. A clear and precise summary of all the information, as well as your calculations.
Name:__________________________
Translating the Situation
Write down any wording or phrasing that will influence the amount of croissants that will be
produced. Include any symbols associated with the information.
Language from problem
Meaning and symbols
Tracking information:
Types of croissants: _____________________ & ____________________
Total dough needed:________________
Total Chocolate needed:_____________
Total prep time needed:_____________
Space for total numbers of croissants…how many? __________________
Name:___________________________
Oven Space Handout
How many of each type do we make each day?
Once you have read the context description, you will have enough information to begin helping
the Miettes to figure out how much money they can make. We need to establish how many
dozens of each type of croissant they need to bake to maximize their profit.
The start, the Miettes have described the total amount of space available for cooking and
cooling the croissants in racks.
1.
To start we need to assign variables to the two undefined quantities of croissants,
remembering we are talking about the baked goods in terms of dozens. We need to
consider plain and chocolate croissants using precise descriptions so…
Let X = ______________________________________________
Let Y = ______________________________________________
2.
Reread the problem and write down the number of dozen of croissants the Miettes
could bake each day if they produced their maximum.
3.
Formulate an inequation representing this situation keeping in mind that they might
not always use the whole oven just for these two types of croissants, but rather a
maximum of that number.
4.
Verify (algebraically) or test to see if the following numbers of dozens of croissants
respect the oven space as formulated above.
A) (50,90)
B) (270, 100)
C) (200.100)
5.
6.
Graph this inequality below. Label the axes and points accordingly. Show your work
here. (table of values and test point)
Label the three points from the previous page on the graph.
7.
Which region shows the number of croissants that will respect the constraint? That is,
where on the graph are the possible numbers of each type of croissant?
______________________________________________________________________________
Name:_____________________
Prep Time at La Miette Handout
Other than physical space limiting the number of croissants the Miettes can make, time will also
restrict the number of each type they can produce. There is a limit on the number of man
hours available, and each type of croissant takes a certain amount of that time to prepare.
1.
Once again, X and Y will represent the number of dozen of each type of croissant…list
those variables again:
● Let X = __________________________ ,
Let Y = ___________________________
2. From the context page, identify how much time each type takes as well as the total time
available…list the information below, and then formulate a constraint.
● Time to produce a dozen plain in minutes:__________________ in
hours:_______________
● Time to produce a dozen chocolate in minutes:_________________ in
hours:______________
● Total time available : _________________
We will use the prep time in terms of hours.
3.
Formulate the constraint that expresses the number of dozen croissants of each type as
they relate to the total time available:
4.
Represent the solution set for this constraint on the graph on the next page. Make sure
to label the graph appropriately. Show the table of values and the verification point
here.
4.
Continued
5.
Select 3 possible pairs (X,Y) from the graph and verify them using the equation. One
point should be from the feasible region, one from the line itself, and one from the non feasible region. Show your calculations and conclude true or false for each.
Name:______________________
Available Ingredients Handout
The next major limit on how many croissants of each type we can produce has to do with
available ingredients. There is only so much dough to use for the croissants, and there is only
so much chocolate available for the type that uses it. Therefore, the amount of croissant
ingredients will restrict the quantity produced t.
1.
Once again, it is important to recall what the variables are for this situation. The
quantity of plain and chocolate croissants (in dozens) needs to be clearly identified.
Let X = _____________________________________
Let Y = _____________________________________
2. Look back at the context page and record the information regarding available ingredients.
Start with the dough, and remember the same dough is used for both types.
● How much dough is used to produce a plain? ________________
● How much dough is used to produce a chocolate?_______________
● How much dough is available in total to use as a maximum? ___________
3.
What constraint can be formulated in order to understand the possible quantities of
croissants based on dough?
______________________________
There are other ingredients!!!
We also know that the Miettes can produce a certain amount of home-made chocolate to fill
one type of croissant. This represents a second, equally important limit, but pertains to only
one type of croissant!
1.
Keeping in mind your variables (especially the relevant one here)
● How much chocolate is used to make one dozen croissants? ________________
● How much chocolate do they have in total? __________________
2.
What other rule can you establish for ingredients. Remember that only one type of
croissant uses this chocolate!
3.
Represent both constraints on the same graph on the next page. Show your work here,
and clearly label your graph. Make sure to highlight or emphasize the feasible region.
3.
continued - Graph:
4.
Pick one ordered pair representing one possible combination of croissants and VERIFY
that point for both inequations below…show your work!
Name:_____________________
Demand Constraint Handout
1.
Which variable is affected by demand? ___________________________
2.
What inequation can be used to express this limit?___________________________
3.
Represent this limit on the graph below:
Name:______________________
Establishing the profit function
Now that we have explored the limits to the Miette's production, we need to take a moment to
consider how they have established their pricing. Read the context again, but this time focus
your attention on information relating to what the Miettes charge for each of their two types of
croissants, as well as what it costs to produce each croissant. Record this data below.
1. Costs:
Cost to make a dozen plain croissants is:________________
Cost to make a dozen chocolate croissants is:_______________
2. Revenue:
A dozen plain croissants brings in ____________ in revenue.
A dozen chocolate croissants brings in _____________ in revenue.
3. Profit:
Since profit is what is left over once we subtract costs from revenue, we can calculate the profit
for each of the two variables with that simple calculation.
What is the profit for a dozen plain?________________
What is the profit for a dozen chocolate?________________
4. Put these two pieces of information together by formulating a rule for the target objective
of maximizing the Miette's profits in terms of X and Y!!!!
P = ________________________________
Name:____________________
Putting it all together!!!
So now you have completely analysed the business model of the Miettes' croissant initiative.
Now is the time to put all of your information in one place and to integrate your work.
1. Identify the variables (again)
Let X = ____________________________ , Let Y = ______________________________
2. List all of the constraints.
3. Place all constraints on the same graph on the next page.
4. Show the final feasible region or polygon of constraints using a bold outline.
5. Locate the vertices of the feasible region.
6. Name 3 other possible solutions from inside the feasible region
1. _______________
2. _______________
3._________________
7. Put all the corner points as well as the others you named into the table using the profit
function.
Vertex
P=___________________________________
8. What is the Miette's maximum possible profit from these points?
__________________________________
9. How many of each type of croissants should they be producing in order to achieve that
profit?
________________________________
REFLECTION
Consider the questions below and write up your responses using a google doc.
1. What did you notice about points at the corners of the polygon compared
to points inside the feasible region?
2. Why wouldn’t the family just make one million croissants of each type in
order to become rich overnight? Give at least 3 reasons from the context.
3. What was the most difficult part of this process?
4. What do you feel you learned in terms of concepts or skills having done this
exploration?
Extension
Design Your Own Business
1
23
4
5
1
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<http://www.vatp.lv/sites/default/files/imagecache/story_full/lampinas_tulpes.jpg>
3
<http://takecontrolmaryland.com/pepco/images/small_business_main_image.jpg>
4
<http://www.glassdoor.com/blog/wp-content/uploads/small-biz1.jpg>
5
<http://www.glassdoor.com/blog/wp-content/uploads/small-biz1.jpg>
2
Name:_____________________
Smart Business!
In order to apply the concepts we learned from the last exploration, it is time to
develop your own business model. You will use the Miettes as inspiration and
work with a partner to create a realistically based situation.
Goal: Your goal is to write up a basic business plan in order to procure a bank
loan. To do so, you must prepare a document explaining the business, its
location, its costs, and its profitability based on real world research! The bank
loan requested should be enough to get you through your first 2 months of
operation. The process is outlined below.
1.
2.
3.
4.
5.
6.
7.
8.
9.
To begin with you will pick a partner. Your choice must be approved by
me.
You must discuss with your partner what kind of business you will be
designing. What 2 products will you be selling? Please remember these
are products to be produced and not services. Keep it realistic!
Company name.
Where will you be located?
What will your business hours be?
Create a Logo for your store sign. This will receive a grade for creativity
and visual appeal.
Select a number that realistically represents the maximum inventory you
could have on hand. The number should be realistic. How will you
estimate this accurately? Show your calculations!
Formulate a constraint based on this number. The constraint should
resemble the one the Miettes used for oven space.
Consider the cost to produce each of your two items. How much would
each cost you to buy/make, and how much money do you have to start
10.
your business? Run these numbers by a banker (me) if you need guidance.
Again, keep it real and be prepared to justify your estimate
Formulate a constraint that represents your budget and costs.
11.
Let’s assume you sell a minimum of one type of these two products. What
would a reasonable estimate be for that product?
12. Formulate a constraint based on your estimate above.
13. How much will you charge for each item? What will your profit be for
each?
14. What rent will you be charged? (How can you get an accurate estimate?)
and can you cite it?
15.
What RULE could be used to calculate the profit of your business?
16. Represent your constraints on a graph.
17. Optimize!
Points to consider:
1.
2.
3.
4.
●
●
●
●
Be Real!!! Where will you learn your information? The information
surrounding space, rent, and pricing can and should be researched
accurately!!! While estimates are acceptable, real numbers found with
research or through demonstrable calculations are better!
Be Neat!! Your information poster should look awesome. If you are not an
artists, clip art and on-line images(cited) can go a long way.
Use Desmos! This will help you tweak your numbers so that a feasible
region will exist. Track and record changes you have to make to your
original numbers to make them work.
IB students must write a short reflection on the process which will include
summaries of the following:
research
challenges
changes made to make the situation viable.
General impressions of the project.
Business Project Template & Checklist
Your business plan should follow the following sequence.
Template - Fill in the following template in order to make sure you have
completed all aspects of the project and presentation.
A) Name 1 - __________________
Name 2 - ____________________
B) Business Name : __________________________________________
C) Two Products sold (with discription and images):
_________________________
&
________________________
D) Actual Location and Address
E) Hours of Operation
F) Logo (created ones show more originality) as cover page:
G) State and title the three constraints you have to formulate. Explain any
calculations used to arrive at the numbers you used. Title each.
H) Show the graph of your polygon of constraint.
I) Make an optimization table.
J) Show your projected profit, explain total costs and ask for the need amount of
money to cover the costs of running the business for the first 2 months.
5
4
3
2
1
Labelling of
Variables
Always defines
both variables
correctly. Answers
all questions
pertaining to
variables correctly.
Defines both
variables correctly
most of the time.
Answers most
questions
pertaining to the
variables correctly.
Defines both
variables correctly
some of the time.
Answers some
questions pertaining
to variables
correctly.
Rarely
defines both
variables
correctly.
Rarely
answers
questions
pertaining to
variables
correctly.
Very rarely
defines
variables
correctly.
Very rarely
answers any
questions
pertaining
variables
correctly.
Formulating
Constraints
Always uses
proper inequation
symbol. Always
formulates
constraints
correctly.
Uses appropriate
inequation
symbol and
formulates
constraints
correctly most of
the time.
Uses proper
inequation symbol
some of the time.
Formulates
constraints correctly
some of the time.
Rarely uses
appropriate
inequation
symbol.
Rarely
formulates
constraints
correctly.
Very rarelyr
uses correct
inequation
symbol. Very
rarely
formulates
constraints
correctly.
Graphic
Representation
s
All graphs are
properly created: A
correct table of
values with at least
two coordinates is
created. Points are
correctly plotted
and the line is
correctly drawn,
while respecting
the inequation
symbol (dotted vs.
solid).
Graphs are
correctly created
most of the time. A
table of values with
at least two
coordinates is
created, these
points are properly
plotted and the
inequation
symbol is
respected when
the line is drawn
(dotted vs. solid).
Graphs are correctly
created some of the
time. A tables of
values with at least
two coordinates is
constructed, these
points are plotted
and the inequation
symbol is
sometimes
respected when the
line is drawn (dotted
vs. solid).
Graphs are
often
incorrectly
created.Table
s of values
contain
several
mistakes,
points are
rarely plotted
correctly and
inequation
symbol is
rarely
respected
when line is
drawn.
Graphs are
most often
incorrectly
created.Table
of values are
either
completely
incorrect or
not present,
points
are incorrectl
y plotted, or
not plotted at
all and
inequation
symbol is
very rarely
respected
when line is
drawn.
Verification
All verifications are
done correctly: a
test point is clearly
indicated (on and
off graph), where
this point is in
relation to the line
is
stated(above/below
), it is correctly
tested and as a
result, the feasible
region is correctly
shaded.
Most verifications
are done correctly:
a test point is
indicated, where
this point is in
relation to the line
is
stated(above/below
), it is correctly
tested and the
feasible region is
shaded.
Some verifications
are done correctly:
a test point is
indicated,, it is
correctly tested and
, the feasible region
is shaded.
Verifications
are rarely
done
correctly: no
test point is
indicated or
tested, but
the feasible
region is
shaded either
correctly or
incorrectly.
The majority
of the
verifications
are done
incorrectly.
No test point
is mentioned
and the
feasible
regions are
incorrectly
indicated.
Optimizing
Optimization is
Majority of
Optimization is
The majority
Optimization
Clarity
done correctly: the
feasible region is
clearly indicated,
corner point are
labelled, 3 points
from the feasible
region are
indicated, the
target objective is
identified, the
optimizing function
is correct and
appropriately used,
the final solution is
correct.
optimization
correct. Feasible
region, points from
the region and
corner points are
indicated, target
objective is
indicated,
optimizing function
is mostly correct
and used with
minimal error.
partially correct:
feasible region,
points from the
region, corner
points, target
objective and
optimizing function
are partially correct.
of the
optimization
process is
incorrect. The
majority of
the following
are incorrect:
feasible
region, points
from region,
corner points,
target
objective,
optimizing
function and
its use.
is incorrect or
incomplete.
All work and steps
are clear,
titled/labelled and
done in the
appropriate place.
Work is neat,
organized, legible,
and follows a
logical order.
Graphs are all
visual, aesthetically
pleasing, all the
components are
labelled and lines
were drawn using a
ruler. .
Works is clear,
legible and done in
the appropriate
place. Graphs were
drawn using a ruler
and components
are labeled.
Work is legible and
done in the
appropriate space.
Graphs are
somewhat clear and
most
components labelle
d. Line was drawn
using a ruler,
Work is done
in the
appropriate
space.
Graphs are
rather unclear
and most
components
have not
been
labelled. A
ruler was not
used to draw
the line.
Work is not
done in the
appropriate
space, graph
is unclear, no
ruler was
used and
there is
minimal
labelling.