Treasure Hunts

Calculus and Advertising

Introduction

Calculus can be used to determine the potential profitability of an advertising campaign. Your task is to use calculus to decide how much money to spend on an advertising campaign to maximize profitability.

Guiding Question

How much money is the right amount to spend on an advertising campaign for a new product?

Questions

1. Choose a company and one of its products. Do research to determine how much the company spends on advertising costs for that product and how many units of that product are sold. Also find out the selling price per unit.
2. Suppose this company is introducing a new product. The company's salespeople want to predict U, the number of units per day they will sell as a function of t, the number of days since the product was introduced. One of the parameters is the amount per day spent on advertising. Here are some assumptions you may make about future sales.

  • The dependent variable is U; the independent variable is t, in days.
  • They spend a fixed amount, M, in dollars per day, to advertise this product.
  • Part of M, an amount proportional to U, is used to maintain present users of this product.
  • The rate of change of U, dU/dt, is directly proportional to the rest of M.
  • Using your data from your research and the assumptions above, find an equation for U as a function of t.
    •  3. Then show the effect of spending various amounts, M, on advertising. You might include such information as whether sales will continue to go up without bound or will eventually level off.
    4. Find out how long it will take the product to start making a profit.
    5. Repeat this project for a different company and/or product.

    Internet Resources

    Companies
    Applied Calculus
    More Calculus
    Advertising
    Calculus
    "Ask" for Products
    Yahoo! Products
    Target for Products
    Safeway
    Costco

    Activities

    Your write-up must include:

    1. The Objective: The main question you are trying to answer with this project.
    2. Bibliography: List of all resources used.
    3. Data: Information about the company, its product, sales volumes, advertising costs, sales revenues, and pricing.
    4. Procedures: Description of what you did in this project and methods used to answer the questions posed.
    5. Visuals: Graphs of the functions developed in this project with labels and captions.
    6. Calculations: Calculations and written explanation of the calculations.
    7.Conclusions: Your conclusions.

    Jean Nanjo, Teacher, and Mitchell Yang, AP student, collaborated to develop this web hunt.

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