**Term Project Option A – MA171 Finite Math**

**Statistical Modeling for Business**

**Purpose:** Analyze and interpret a basic business model for manufacturing and selling a product.

TABLE 1: Business Modeling Functions | ||

Function | Formula | Definition |

Price-Demand | p(x) = m – nx | Expected number of items sold at price $p |

Cost | C(x) = a + bx | Total cost = fixed costs plus cost per item x |

Revenue | R(x) = xp | Items sold at price $p |

= x(m – nx) | Price determined by price/demand function | |

Profit | P(x) = R(x) – C(x) | Profit is Revenue minus Costs |

Background: Your Company (Parkville Motor-Works Sales) makes electric mini-bikes. Your boss assigned your team the task of determining the price/demand, cost function, break-even and profit loss analysis. Your team hits the ground running and collects the following data. (Being the team leader: You may not know how they did it…just be happy it got done)

Table 2: Price/Demand | |

x (thousands) | p($) |

7 | 550 |

13 | 375 |

19 | 300 |

25 | 150 |

Table 3: Cost | |

x (thousands) | Costs (thousands $) |

5 | 2,450 |

10 | 2,950 |

20 | 3,750 |

30 | 4,900 |

*Now comes the fun part!* **Analysis: **

- Build a model of the Price/Demand
- Graph the points and fit a curve to the points.
- Interpret the graph (what is it telling you?). Does the graph look linear?
- Use
*Linear Regression*to get the Price/Demand equation (what are the values for m and n?)

- Build a model for Cost
- Graph the points and fit a curve to the points.
- Interpret the graph (what is it telling you?). Does the graph look linear?
- Use
*Linear Regression*to get the Cost equation (what are the values for a and b?)

- Do a TLAR and ASSUME the best price should be $360.
- Graph the Cost Function and the Revenue function [R(x) = 360x]
- Where is the Break-even point? (i.e. where do the two lines cross?)

**But…you know better than to ASSUME!**

- Build the actual Revenue model
- Graph the Revenue equation [R(x) = x(m – nx)]
- Interpret the graph (what is it telling you?)

- Break-even and Profit-Loss
- Graph the Revenue function and the Cost Function on the same graph
- Where is the break-even point (i.e. where do the two functions cross?)

- Graph the Profit function.
- At what output does your company break even?
- At what output and price will the maximum profit occur?
- Could you conclude: Maximum Output always mean Maximum Profit?

**MA171 Project Option B**

The PTO of Parksville Elementary School is having a Bazaar and all of the classes are expected to contribute to this fun and exciting event. A group of third grade teachers, and some dedicated parents, decide to sell cotton candy.

###### A little research shows you can get the machine for $50 and cleaning supplies will cost $10. The sugar and cones come in a package for $10, enough to serve 100 (you can get a refund for unopened packages). The committee thinks they could probably sell 150 of these sticky globs of goodness, so they order two packages of sugar and cones.

One of the parents asks: If we sell the cotton candy for $1.50 each, how many servings must be sold to break even? How much needs to be sold so we can buy a $340 computer package for our science lab? You have volunteered to create the equations and graph the cost analysis chart so you will be able to answer that question as well as these:

- What are the Cost equations? Graph the equation.
- What is the Revenue equation? Graph this equation.
- Where do these two lines cross? (Break-even point)
- Profit equals Revenue minus Costs (P = R – C). What are the Profit equations?
- How much profit will be made if 99 servings are sold?
- How much profit will be made if 102 servings are sold?
- Do you always make more profit by selling more? Why?
- How much profit will be made if 150 servings are sold?
- How many servings must be sold to make $340? Can you do it?
- How much would you have to sell each serving for to make a profit of $340 without getting extra supplies?

##### You must explain all assumptions you make…

**ROE (Rules of Engagement)**

- Neatness counts!
- Keep it simple

– technical is impressive (only to other technicians)

- most people are afraid of math (tune out as soon as they see an equation, graphs are better)
- know your audience…if presented at the 9
^{th}grade level, almost everyone will understand

- Charts must be explained

- clearly mark your coordinate axis
- proper scale for your charts

- Answer the questions asked

- explain/identify what the question is
- keep technical stuff separate (it may obscure the answer (see #2 above))

**Grade is based on:**

20% Neatness, 20%Organization, 20%Correctness, 20%Charts/Graphs, 20% Presentation

**MA171 Project Option C**

So…my wife and daughter decide to go shopping and they just informed my that I am expected

to watch Jason; my 7-year-old grandson. Now…as a matter of fact…I don’t recall volunteering

to babysit…but…um…it beats going shopping for shoes, jeans, or a purse with my wife and

daughter.

Anyway…what can one do with a 7-year-old? Oh, I know…I’ve been meaning to count (and roll) all the coins I have in my mason jar. With Jason’s help, we can do this easy…and grandpa can engage in some quality time.

So, we start sorting and counting…when all is said and done…we end up with a bunch of quarters, a whole lot of dimes, a good number of nickels, and a fair quantity of pennies.

Jason counts the coins and comes up with 228. We calculate they are worth $19.90. In addition, Jason notices the number of nickels is equal to the number of quarters plus 2. He is really good with math…some things may be genetic…and also noticed the number of pennies is equal to the number of dimes plus half the number of quarters.

Assignment:

- Generate (and solve) the linear equations for this problem,
- State the number of quarters, dimes, nickels, and pennies in the jar.
- As an incentive, I tell Jason he can keep any of the coins that are remaining from a full roll…now you know why I am his favorite grandpa. FYI: Quarters are stacked 40 to a roll ($10.00), dimes are stacked 50 to a roll ($5.00), nickels are stacked 40 to a roll ($2.00), and pennies are stacked 50 to a roll (50¢). So, how much money did Jason get to keep?

**ROE (Rules of Engagement)**

- Neatness counts!
- Keep it simple

– technical is impressive (only to other technicians)

- most people are afraid of math (tune out as soon as they see an equation, graphs are better)
- know your audience…if presented at the 9
^{th}grade level, almost everyone will understand

- Charts must be explained

- clearly mark your coordinate axis
- proper scale for your charts

- Answer the questions asked

- explain/identify what the question is
- keep technical stuff separate (it may obscure the answer (see #2 above))

** **

**Grade is based on:**

20% Neatness, 20%Organization, 20%Correctness, 20%Charts/Graphs, 20% Presentation

**MA171 Project** Option D

The PTO of Parksville Elementary School is having a Bazaar, and all are expected to contribute to this fun and exciting event. The Kindergarten teachers, along with some dedicated parents (meaning mostly YOU!) decide to sell popcorn; hoping to raise enough money to buy a $300 educational software package for the library. The committee thinks $1.50 is a good price for a bag of popcorn.

One of the parents asks: How many bags must be sold to break even, and how many do we need to sell to purchase the software? Fortunately, you are enrolled in a Finite Math class and can create the equations, do the analysis and answer that question. You volunteer to get them the answers for the next meeting, planned for tomorrow evening. Before leaving, you track down the continuity binder.

Because you are a type ‘A’ person, you can’t wait to get started; but the stores are closed, and you don’t know what the costs are. However, looking through the continuity binder, you find the overall costs from the previous two years. Assuming the costs will be constant this year, you start working on your presentation.

Year | Number sold | Total Cost |

2007 | 80 | $52 |

2008 | 120 | $72 |

- Graph these points. What is the Cost equation?
- What is the Revenue equation? Graph this equation.
- Where do these two lines cross? (Break-even point)
- Profit equals Revenue minus Costs (P = R – C). Simplify the Profit equation?
- How much profit will be made if 100 bags are sold?
- You estimate (at most) 150 bags will be sold. Will you be able to buy the software?
- If you could sell 150 bags: How much should you charge to make enough to buy the software package? Should you raise the price?

##### You must explain all assumptions you make…

**ROE (Rules of Engagement)**

- Neatness counts!
- Keep it simple

– usually, technical stuff is only impressive to technicians

— keep technical stuff separate (it may obscure the answer)

- most people are afraid of math (tune out as soon as they see an equation, graphs/pictures are better)
- know your audience…if presented at the 9
^{th}grade level, almost everyone will understand

- Charts must be explained

- clearly mark your coordinate axis
- proper scale for your charts

- Answer the questions asked

- Clearly explain/identify what each question is and what the answer is

**Grade is based on:**

20% Neatness, 20%Organization, 20%Correctness, 20%Charts/Graphs, 20% Presentation

**MA171 Project Option E **

You’ve been put in charge of the annual Christmas Party. *(How many times have we told you not to volunteer?)* Anyway…you want to get a good estimate of the costs this year. You search for the Continuity Binder and (naturally) couldn’t find one. However, you know that SSgt Kringle was the POC of the party last year, so you track her down. She wasn’t much help; but, she managed to find the receipts for the total costs from the previous two years: The costs were:

Year | Number attended | Total Cost |

2007 | 52 | $1,030 |

2008 | 122 | $2,880 |

So many people showed up last year because it was the 25th Anniversary of the squadron. However, there is nothing special going on this year so you expect about 100 people will show. Your Squadron Commander was very impressed with the NCO Club last year and you decide to duplicate the events of that night. You contact the NCO Club and determine that the costs this year will be (essentially) the same as last year.

- Graph these points. What is the Cost equation?
- How much should each person be charged in order to break-even?
- If you will invite 10 VIPs (who will not be charged for their meals): How much should you

charge to attend?

- What color carnation should you wear if the décor will be in red and white?

##### You must explain all assumptions you make…

**ROE (Rules of Engagement)**

- Neatness counts!
- Keep it simple

– usually, technical stuff is only impressive to technicians

— keep technical stuff separate (it may obscure the answer)

- most people are afraid of math (tune out as soon as they see an equation, graphs/pictures are better)
- know your audience…if presented at the 9
^{th}grade level, almost everyone will understand

- Charts must be explained

- clearly mark your coordinate axis
- proper scale for your charts

- Answer the questions asked

- Clearly explain/identify what each question is and what the answer is

**Grade is based on:**

20% Neatness, 20%Organization, 20%Correctness, 20%Charts/Graphs, 20% Presentation

**MA171 Project Option F**

** **In Chapter 3 of our *Finite Mathematics* book: Work problem 47 on page 165.

**MA171 Project Option G**

** **In Chapter 5 of our *Finite Mathematics* book: Work problem 16 on page 284.