Please find the attached file here including the project description.

**Math115 Project**

Radomir Pavicevic, Anna Chalkia, Evangelia Nanou

**Part A**

A1: Target Market

We decided that our Target Market will be students and faculty staff of the American College of Thessaloniki (ACT) in Greece. The number of the ACT community is approximately from 500 to 999 students and above 100 people of professors and faculty staff.

A2: Design the Shirt:

Two-sided black and white.

The front logo: Save the planet / The back logo: Be the change you want to see.

The T-shirt.

A3: Marketing Plan:

We decided that we want to sell this kind of T-shirt with the specific topic (Climate Change) because it is a strongly important issue and a current social trend. It has a meaningful goal and it is a very recent problem especially in these years we are living in. We are going to sell this T-shirt in the campus of ACT three days a week, during the break 14:00pm – 16:00, for the Spring 1 semester. Moreover, the advertising techniques we are going to follow are posters all over the campus and digital promotion through the Students Government Association. The goal is to collect money and donate them in ecological and environmental charities to help save our planet from the plague of the climate change.

For instance:

Anna: Monday (2 hours)

Radomir: Wednesday (2 hours)

Evangelia: Friday (2 hours)

Furthermore, we will include 1 hour for each of us for distributing the poster one day a week. (Thursday)

A4: Reachable Target Market:

Our reachable target market will be 250 people. So D=250. This number is being generated from the assumption that during the break many people pass in front of the cafeteria and follow SGA on Instagram. Also, we will be distributing the posters and because it is for a good cause, professors, faculty staff and students will be interested.

A5: Unit Cost:

€5.75+2*€0.55

We will order 125 shirts, and it is two sided with one pattern colour.

A6: Fixed Costs:

The poster.

A7: Polling Plan:

https://docs.google.com/forms/d/e/1FAIpQLSd_5lrLh7b5M-6oGN9Q4Iijq4VS6UhdveRSMbOQR_zSwFNiaw/viewform?usp=sf_link

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Business Calculus – Math 11

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Project: T-Shirt Sales Campaign

In this project, you will design and test the viability of a T-shirt design. The project will be done in five parts:

A. Design a T-shirt and create a marketing plan;

B. Take a poll to test the marketability of the shirt;

C. Use the results to determine the demand function for your T-shirt to obtain the revenue, cost and profit

functions;

D. Use the tools of the course to find a selling price that will maximize revenue and profit.

E. At the conclusion of your project, you will write a report and submit it to the instructor presenting all

your analysis and final results. This report must address all aspects of the project, thus there should be

write-up on parts A – D. The report will be submitted in MS word format, while all the mathematical

analysis should be done in MS Excel or a program of your choice.

Important Points:

• You will work in groups of three which will be formed within the first weeks of the semester.

• Everything you turn in must be typed. The only exception is the design of the shirt and your advertising

material, if any. These must be neatly drawn and colored.

• Make sure to include the name of group members, your section, and your instructor’s name on all work

submitted.

• The MS word report as well as all other electronic files should be submitted through Moodle.

• In addition to the electronic Moodle submission, you should submit all your hand-produced

material and a printout of your final report in a two pocket folder. On the outside of the

folder write the names and relevant contact information (i.e. email, etc.) of all group members.

Deliverables and Deadlines:

Deliverables Dates

Part A See Moodle

Parts B & C See Moodle

Parts D & E See Moodle

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Part A: Design, Marketing and Cost Estimates

A1: Target Market: You first must decide to whom you want to sell the T-shirts; this will be your TARGET

market. Examples include freshman at ACT; business majors; students who play varsity sports, etc. Your

written description must be specific. You will need to have a reasonably accurate estimate of how many people

are in your target market; this number does not need to be exact but it should come from a concrete,

identifiable source. You must settle on a single number that is an estimate of the size of the target market. If

you can’t find a source or a sensible way to estimate the size of your target market then you will have to find a

different target market.

A2: Design the Shirt: Design a T-shirt that will appeal to your target market. Prepare a careful drawing of

your shirt. It does not have to be drawn using computer tools but it must be neat. You will use your drawing

in your pre-marketing poll so the drawing of each side of your T-shirt must be at least 8.5″ by 11″. Write an

explanation in a paragraph or two about why your T-shirt will appeal to your target market, identifying

what makes your T-shirt special or unique. [Think about your sales pitch: why would consumers in your

target market purchase your T-shirt over competing T-shirts?] You also need to include the following specific

information about your shirt: (1) the (background) color of the T-shirt; (2) whether the T-shirt design will be

one-sided or two-sided; (3) a list of ALL colors used on the front of the T-shirt and (4) (if the shirt is two-

sided) a list of ALL colors used on the back of the T-shirt. Note that T-shirt designs and slogans should be in

good taste.

A3: Marketing Plan: In this section you will decide how to market and sell your shirt. You need to identify

where and when you will sell your shirt (be specific: on which days, how long on each day, and how many

days, weeks or months). Also indicate any advertising techniques you plan to use, such as flyers, posters,

online ads, etc. Include a sample of any advertisements.

Include an estimate of the total number of hours that members of your team will spend designing, marketing

and selling the T-shirt. This includes time spent creating advertising materials, passing out and posting flyers,

manning tables, etc. Explain carefully how you get your estimate. Each member of your team can contribute

up to eight hours of unpaid labor to marketing and selling the T-shirt. Thus a group of four is allowed a total

contribution of 32 hours of unpaid labor. Each additional hour that you spend should be added to your fixed

costs at the Greek minimum wage rate of 4.50 euro per hour.

A4: Reachable Target Market: Despite our best intentions, marketing plans do not reach every member in the

target market. You need to estimate the proportion of your target market that you will reach with your

marketing plan. This is called your REACHABLE target market, and is denoted Do.

To do this, think about how much advertisement you will do and how well it is likely to work. For example,

assume that a typical undergraduate student passes through the ACT cafeteria at least once a week between

10am and 2pm. If you have a table in the cafeteria two days a week for two hours between 10am and

2pm, perhaps one fifth of undergraduate students will see it. Whereas if you can distribute fliers to everyone in

your target market, you might reach 90% of your market.

Your reachable target market must be a single number (not a range) and you must include a short explanation

for how you got this number based on your marketing plan. Note that the size of your reachable target

market should be at least 100.

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A5: Unit Cost: Refer to the pricing chart on Shirts ‘R’ Us Pricing Chart to determine the cost of making one

T-shirt. This is your unit cost. Note that the unit cost depends on how many shirts you order. You will get a

precise value for this number later in the project, but for now use half your reachable target market (i.e.

½Do) as your estimate. Make sure to indicate how you determined your unit cost based on the color of your

shirt, whether it is one-sided or two-sided and how many pattern colors you use.

For example, suppose your reachable target market is 350 and you plan to sell a two-sided green T-shirt with

two pattern colors (red & white) on the front and one pattern color (red) on the back. Since Do = 350, you will

order 175 shirts, so your costs are based on the 51-250 line. The shirt is green (i.e. colored) and two-sided

with a total of 3 pattern colors. Therefore each shirt will cost €5.75 + 3 × €0.40 = €6.95. This is your unit cost.

If Do = 750 and you create a one-sided white shirt with 4 pattern colors (green, yellow, black & red), your unit

cost would be €4.20 + 3 × €0.20 = €4.80.

A6: Fixed Costs: Give an estimate of the fixed costs associated with selling your product. This includes all

advertising costs such as labor, printed materials, radio ads, table rentals, permits, security, etc. It does not

include the cost of making the shirts but it does include the cost of hours spent marketing and selling the

T-shirt beyond your group’s quota (see part A3). If you plan to use printed advertising material (posters, fliers

or pamphlets) make sure to determine the actual cost of these items from a supplier such as the ACT

bookstore or other. Note that you CANNOT use your free ACT print quota to produce fliers and pamphlets.

You should have a single number estimate for your fixed costs. It is very unlikely that your fixed costs will

be zero.

A7: Polling Plan: Finally, you need to develop a plan for how you will poll your target market to find out how

people feel about your design. Based on your marketing plan (part A3), decide where and when (give dates

and times) you will conduct this poll. For example, if you plan to sell the shirts on Mondays and Thursdays

from 12-2pm, then you should take the poll on the same days of the week and in the same place where you

intend to sell your shirt. Note that you are only designing a plan at this stage, not conducting the actual poll.

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Shirts ‘R’ Us Pricing Chart

The following chart provides the unit cost (i.e., the price you have to pay for one T-shirt) for 100% cotton

short-sleeve shirts, regardless of shirt size.

The factors that affect the price are: the color of the T-shirt; whether the design is printed on both sides or

only on one side; and how many pattern colors are used in the design.

The shirt can be white or any other color, but it must be a single color.

Since each side of the shirt is printed separately, in a two-sided design you need to add the total number of

pattern colors appearing on each side of the shirt even if the same color is used on both sides.

How to Read the Pricing Chart

• The base color of your T-shirt will determine the column to use.

• If your design is one-sided, then the first pattern color in your design is free – you only need to pay for

any additional pattern colors.

• If your design is two-sided, then you need to pay for each pattern color.

Quantity Ordered

Price per Two-sided Shirt (no pattern colors included)

One-sided Shirt with one pattern color included

Cost per additional

pattern color

White Shirt Colored Shirt

0-50 €5.10 €6.25 €0.55

51-250 €4.60 €5.75 €0.40

251-500 €4.20 €5.25 €0.20

≥ 501 €3.75 €4.75 €0.15

Add €1.50 to the unit cost for a long-sleeved, heavyweight cotton shirt.

Add €1.50 to the unit cost to print on one or both sleeves (but any colors printed on the sleeve

must be one of your pattern colors).

Examples:

1. Your reachable target market is 800. You create a one-sided black shirt with two colors (white and red).

Base your unit cost estimate in A5 on ½D0 = 400 shirts. The shirt is one-sided and colored, so your

base cost is €5.25 which includes one free pattern color. You need to pay for one additional pattern

color at a cost of €0.20, so your cost for each T-shirt is €5.45. This is your unit cost.

2. Your reachable target market is 500. You create a two-sided white shirt with two colors (blue and red) on

the front and three colors (blue, yellow and green) on the back.

Base your unit cost estimate in A5 on ½D0 = 250 shirts. The shirt is two-sided and white, so your base

cost is €4.60 and you have to pay for all pattern colors. You have a total of five colors (note that blue

counts twice) at a cost of €0.40 each, so your unit cost is €4.60 + 5 × €0.40 = €6.60.

3. Your reachable target market is 1200. You want a two-sided white long-sleeved heavyweight shirt with

two colors (red and black) on the front and one color (red) on the back.

Base your unit cost estimate in A5 on ½D0 = 600 shirts. The shirt is two-sided and white, so your base

cost is €3.75. You need to pay for three colors at a cost of €0.15 each. The long-sleeved shirt costs an

additional €1.50, so your unit cost is €3.75 + 3 × €0.15 + €1.50 = €5.70.

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Part B: Target Market Poll

B1: Polling: Now you get to find out how people in your target market feel about your T-shirt design. You

need to poll at least 50 people using the polling plan you developed in A7. Your goal is to determine the

maximum price that people in your target market are willing to pay for your shirt. Work in pairs. Have the

drawings of your T-shirt ready for display and carry clipboards so you look and feel “official”. Record the

responses on your clipboard. Please talk to your instructor if you would like to do an online poll.

• You are not asking the person what the price should be or what this person would recommend you set as

your price, but rather the maximum price that he or she would be willing to pay.

• Do not suggest particular prices. Let the person come up with the price it on his or her own.

• It is best to poll only one person from a group, and make sure to keep people you are polling separate, so

that one person’s opinion doesn’t influence another’s.

• Do not include in your 50 responses someone who does not stop to listen to your sales pitch.

• Someone who listens to your sales pitch but would not want the shirt at any price is recorded as €0.

B2: Responses: Tabulate your results using table similar to the sample below.

• Round your responses to the nearest whole euro (i.e. round €7.50 to €8 and €5.45 to €5).

• Your table starts with the highest price that you were given. Include a row for every whole dollar amount

starting at the highest price down to €0, even if no one gives that price.

• Count the number of responses for each price. Note that some of these may be zero. For example in the

sample below, nobody answered that they would pay €9, so that price has a response of zero.

• Make sure that the total number of responses adds up to the number of people you polled.

B3: Cumulative Responses:

• In the example below, for the price of €7, the value of 12 in the “Cumulative Responses” column is the

sum of the responses at the prices of 7, 8, 9 & 10 (i.e. 4 + 3 + 0 + 5).

• You can also obtain this value by adding the “Cumulative Responses” entry for price level 8 to the

“Responses” entry for price level 7.

• This value gives us the number of people polled who would be willing to pay that price or higher.

B4: Demand: The demand is obtained by scaling the results of your poll to your reachable target market.

• To do this you need to scale your “Cumulative Responses” column based on the value of D0.

• To determine the scale factor, divide D0 by the number of people polled.

• Multiply each entry in the “Cumulative Responses” column by the scale factor to obtain the

corresponding entry in the “Demand” column.

• In this example, the reachable target market is D0 = 350, and a total of n = 50 people were polled.

Therefore, the scale factor is 350 ÷ 50 = 7. The “Demand” values are obtained by multiplying the

“Cumulative Responses” values by 7.

Demand Based on Poll Results (D0 = 350 and n = 50)

Price (€) Responses Cumulative Responses Demand

10 5 5 35

9 0 5 35

8 3 8 56

7 4 12 84

6 5 17 119

5 12 29 203

4 8 37 259

3 4 41 287

2 5 46 322

1 2 48 336

0 2 50 350

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B5: Polling Experience: Briefly discuss your experiences conducting the poll. Did people listen to your sales

pitch? Were they receptive to your product? If you were to do this again, would you revise your marketing

plan or change your polling strategy? Include some specific comments or reactions of the students you polled.

B6: Scatter Plot: Create a graph in Excel that shows the scaled demand for your T-shirt versus the price people

are willing to pay.

• Plot the price (in €) on the horizontal axis, with the price increasing from €0 to the highest price, and

Demand (number of responses) on the vertical axis. Make sure to label the axes appropriately.

• Use Excel to create your graph. Do not draw it by hand.

• There should be one point for every price from €0 to the highest price given.

• You should get a downward sloping pattern. If you don’t, there’s something wrong.

• Do not connect the dots or bars.

NOTE: Putting price on the horizontal axis is the mathematician’s convention. Make the graph this way, even if

you have seen the demand on the horizontal axis (this is the economist’s convention).

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Part C: The Demand, Revenue and Profit Functions

For the demand, revenue, cost and profit functions below, make sure to write your models with

ALL the decimal places the computer gives. Do NOT round the numbers. Write out the

formula in full and do not use any abbreviations.

C1: Model the Demand Function D(x): Based on your graph of price versus demand from part B6, decide on the

best model for your demand. It can be linear or quadratic or cubic or exponential or logistic. Plot a few trend

lines in MS Excel and choose the one with the highest R2 value as shown in class.

C2: Revenue Function R(x): Use your demand function from C1 to determine your revenue function based on the

formula below.

R(x) = (selling price) × (number sold) = (selling price) × (demand) = x . D(x)

C3: Cost Function C(x): Use your demand function from C1 to determine your total cost function. The unit and

fixed costs are those you determined in part A.

C(x) = variable cost + fixed cost = (unit cost) × (number sold) + (fixed costs)

= (unit cost) × D(x) + (fixed costs)

C4: Profit Function P(x): Use your revenue and cost functions to determine your profit function.

P(x) = R(x) – C(x)

Without rounding the values, simplify your profit function as much as possible for readability, as it

will be an important part of your presentation. Use parentheses as needed.

The next step is very important and will determine whether you can complete part D of the project. When you

have written down your models for demand, revenue, cost and profit, use Excel to graph R(x) and P(x) over

the interval of prices that people are willing to pay for your T-shirt. Perform a visual check that the graphs

of R(x) and P(x) each reach a maximum value inside this interval. If one of these graphs does not reach a

maximum inside the interval, check your formulas and if necessary choose a different model for demand.

Choosing a new model will be done by your instructor, after you arrange a visit in his office during his office

hours.

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Part D: Optimization

In order to complete this part your formulas for D(x), C(x), R(x) and P(x) from part C must be correct and

approved by your instructor. You will use these formulas to answer the questions below.

D1: Maximize Revenue and Profit: Find the price that maximizes revenue using Excel and the calculus

concepts learned in class. You will then enter these values into the REVENUE and PROFIT tables below.

The “Maximize Revenue” and “Maximize Profit” tables below have rows for “Math Results” and “Business

Results”.

To get the “math results” price, round the price you found using Excel to the nearest cent. Use this

value for x in your formulas from Part C to determine the values for the demand, revenue, cost

and profit. Round the demand to the nearest whole number and all dollar values to the nearest

cent.

“Maximize Revenue”

The price (€) that maximizes revenue is x = (give all the decimal places for x)

Price (€) Demand (Shirts) Revenue (€) Cost (€) Profit (€)

Math Results

“Maximize Profit”

The price (€) that maximizes profit is x = (give all the decimal places for x)

Price (€) Demand (Shirts) Revenue (€) Cost (€) Profit (€)

Math Results

D2: Give a short answer to the following question: Why is the price that maximizes revenue different from

the price that maximizes profit?

D3: At what rate is the profit changing at the price that maximizes the revenue. Give your answer with

units and show how you determined this.

D4: Considering the possible prices from the tables in Part D1 (or some other price), which price (be specific)

would you sell your T-shirt for? Make sure to explain your reasoning.

D5: Based on the price you gave in D4, what is the demand for your T-shirt? In part A of the project, you

used an estimate for the demand to obtain your unit cost. Calculate your new unit cost based on the demand

that you just found and show your work.

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D6: Suppose that you ordered and paid for exactly enough T-shirts for half of your reachable target market

before you polled students to find your demand function D(x). If D(x) is the same as in part C, write a formula

for the profit function P(x) for this situation. (Don’t forget to include the fixed costs from part A.)

D7: Suppose that you tripled your demand function D(x) from part C, i.e., you used 3.D(x) in all your

calculations (beginning in part C) in place of D(x). What is the price (rounded to the nearest cent) that

maximizes the new REVENUE function? What is the new maximum REVENUE (rounded to the nearest

cent)? Explain your answers. You don’t have to fill in a new table.

D8: Suppose you added €150 to your fixed costs (beginning in part C), and did all your calculations again.

What is the price (rounded to the nearest cent) that maximizes the new PROFIT function? What is the new

maximum PROFIT (rounded to the nearest cent)? Explain your answers. You don’t have to fill in a new table.

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Part E: Final Report

Prepare a project report in MS word.

Checklist for points to discuss in your report.

• Design and appeal of T shirt

• Target market (description, size, source)

• Marketing plan with sample flyers and other advertising materials

• Reachable target market size

• Estimated unit and fixed costs

• Where, when, and how poll was conducted

• Table showing results of the poll

• Graph of price versus demand

• Demand, Revenue, Cost, Profit functions (with rounded coefficients)

• Optimization procedure and tables (Math Results, Business Results, etc.)

• Final price recommendations with explanation

Each member of the project group must be prepared to answer questions on any aspect of the project!