Mesquite Trees, Anon.

Thorn Avenue

Lubbock, TX 79414

Dear Calculus Student,

You don't know me - my name is Warren Flubberneck, but you may remember my neighbor, Betty. She related to me how helpful and resourceful you were on a project she placed across your desk last semester.

On the recommendation of a number of other companies with which we are affiliated, we have would like to contract with you to resolve a cost-issue which pertains to the recently announced expansion of our world-wide headquarters in Fort Stocken. While the building expansion is to be connected to the existing structure, recent regulations passed by the Fort Stocken city council require us to run a new line from the local power hub to provide electronic connectivity and power. Unfortunately, this necessitates running conduit through the wetlands behind the new construction, which makes it difficult to determine the optimum path along which the conduit should be laid. (See Project Map below).

We estimate that it will cost approximately 2 1/2 times as much to run conduit through the wetland section of the path as it does through non-wetland ground. Owing to concessions made by Fort Stocken to encourage business development, we only have to run conduit from the edge of our building to a point at which the power company will make power available, as shown on the map which follows.

We need to know what optimal route to select for laying the conduit to minimize our costs. We will eventually obtain a firm estimate for the cost increase for running conduit through the wetland ground. Would you provide to us several models for optimal conduit routing assuming the following fixed scenarios:

Base Cost (per foot) | Wetlands Increase Factor |

$8 | 1.7 |

$8 | 2.2 |

$8 | 2.3 |

$8 | 2.8 |

We need both a planned route for each scenario and a cost estimate.

We look forward to receiving your analysis of this problem. The
final 3--5 page report should be submitted by the
**29**th of **January**. As is our policy, we
have arranged that a qualified (if erratic) mathematician in the area,
the estimable Dr. Pearce, be available to answer any
questions you may have in the course of your work.

Yours sincerely,

Warren Flubberneck, President

MPC, Inc.

Project Map...

All reports submitted to MPC, Inc. should be written so that the forewomen and foremen of the construction unit implementing the report can understand and apply the information contained therein. Owing to the preeminent position of MPC, Inc. in the field, all of our forepeople have degrees in 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.