# Math 106--Calculus II: Project 3

## ...the continued cost of expansion

### by Gavin LaRose (glarose@umich.edu), Nebraska Wesleyan University, November 1995

permission granted to use and distribute free in an academic setting

Success is yours, and the contracts are pouring in! After two resoundingly successful contracts with large companies (both, curiously, related to the giant MPC, Inc. conglomerate), you have received a third such contract. The letter you received from the MPC, Inc. conglomerate comes on the heels of their use of your previous correspondence with regards to the expansion of their world-wide headquarters, and follows.

## The letter...

### M.P.C., Inc.

1 MPC Drive
Lonlinc, SK 04685

5 Sept., 1995

Sophisticated Technical Contractors, Inc.
Lonlinc, SK 04685

Dear STC:

We were favorably impressed by your cost analysis for the different expansion proposals which we were considering earlier this Fall, and so have contracted with you for additional analysis as we move in the construction phase of the expansion.

Because of recent legislation passed by the Lonlinc city board, we are having to lay 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, and calculating the cost of laying it therefore becomes complicated by the fact that the expense is proportional to the amount of water saturating the soil. We therefore need from you an accurate estimate of the amount of water in the soil that we will have to dig through to lay the conduit.

To determine the amount of water along the path that the conduit will be laid our in-house engineers took measurements at intervals along the path; their results are summarized in Table 1, below.

 x = (in yds) % H2O = x = (in yds) % H2O = 0 1 9 12 18 21 10.0% 10.5% 16.0% 17.5% 23.5% 24.0% 33 34 45 47 58 60 25.0% 26.5% 42.0% 40.0% 19.5% 17.0%

Table 1: Percent water saturation in ground along conduit path

The ditch to be dug for the conduit is shown schematically in Figure 1.

Figure 1: Ditch and conduit configuration.

In as much as we hope to have a full budget outline for the expansion project finished by the end of the calendar year, we need your estimate for the the cost of the ditch (you are to assume that the cost of the ditch, per cubic foot of water in the soil, is \$C) by the 15th of December at the very latest. If you have questions as you are working on the project, you should consult with chief scientist, Dr. Gavin LaRose, who has been instructed to assist you insofar as may be possible. Please also note that you should be in touch with him at the ends of the weeks of the 27th of November and 4th of December to update us on your progress. As is usual, we have enclosed a copy of our technical report specifications, which should govern the presentation of your results.

Yours sincerely,
Warren Fubbet, President
MPC, Inc.

Encl: Technical report requirements

## The technical specifications...

### M.P.C., Inc.

M.P.C., Inc. Technical Report Requirements

All reports submitted to MPC, Inc. should be written so that all members of the board of directors can understand the issues raised therein and therefore be able to appropriately use the material contained therein. The board members can all boast of a good university education including in excess of a year of mathematics -- of which they probably remember only about one semester of Calculus.

Reports should further:

• Be written in the first person plural (e.g., "We found the requisite data from the map...").
• Include mathematical formulas in the the text of the body of the report as appropriate to describe the methods and results obtained.
• 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.

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