Mesquite Trees, Anon.
Thorn Avenue
Lubbock, TX 79414

Dear Calculus Student,

I am writing to ask for your assistance in a matter of great importance to our small but reputable poster printing company.

We recently received an order from a most valued corporate customer, a florist called The Petal Peddler. Some of the details of the order were lost in an unpleasant paper-shredder accident involving Bud, our receptionist (he is recovering nicely, but the order form was not so fortunate). After regaining consciousness, Bud was able to remember that the florist wants a poster featuring a stylized flower, in the shape of the well-known polar graph, the rose curve r = cos (n * theta), scaled to fit in a circle of radius one meter. The flower is to be printed in solid purple (filling the petals) onto otherwise blank card stock. Unfortunately, Bud was unable to remember the value of n.

We have, of course, asked a discreet employee of The Petal Peddler to find out the value of for us, but by the time we receive this information, it will be too late to order the ink we need to process the order. Here's where we thought you might be able to help us out, having heard of your legendary expertise in the field of polar functions. The rather costly purple ink we use comes in containers, each with enough ink to cover 100 square meters. Not knowing the value of n, we do not know how many containers to buy in order to prepared for this 8000-poster printing job. Could you please give us some advice as to how many containers of ink to buy in order to (a) be sure that we will be able to fulfill our obligations to the florist, and (b) keep from wasting too much of this expensive ink? If it is impossible to advise us without knowing the value of n, please share any relevant findings with us so that we can best prepare for the embarrassing moment when we will have to confess our mistake.

I would appreciate your report by Tuesday, 23 Apr 2002, at 5:00 pm, since my ink order must be faxed to my supplier by noon the next day. Thank you in advance for your assistance.

Sincerely,

Art C. Guy

Plant Manager
Ink, Inc.

cc: Dr. Kent Pearce

All reports submitted to Ink, Inc. should be written so that the directors receiving the report can understand and apply the information contained therein. Owing to Ink, Inc. preeminent position in the field all of our managers have degrees in industrial engineering, and thus have had college level mathematics, including calculus---unfortunately, however, their long experience in the field precludes a ready knowledge of the same. Therefore, the reports should assume a strong precalculus and basic calculus (about half a semester of calculus I) background, but should not expect a knowledge of much more than that.

Reports should further:

• Be written in the first person plural (e.g., "We found the requisite data from the figure...").
• Include mathematical formulas and appropriate graphs in the body of the report as appropriate to describe the methods and results obtained. (While the report must be typewritten, it is fine to neatly hand-write formulas if that significantly simplifies its generation.)
• Clearly explain how the mathematical formulas that are included bear on the problem being solved.
• Consist of:
  • An Introduction, describing the problem to be solved, and an indication of the mathematical method used to solve it.
  • A Body, describing the mathematical problem that was solved to answer the question(s) posed in the introduction, and the solution to it.
  • A Conclusion, summarizing the results obtained from the solution described in the body and clearly stating their relevance to the original problem as described in the introduction.
  • Be 2.5--5 pages in length, excluding supporting figures and diagrams in an attached appendix.