MATH 1080-001 Optimization Writing Project (n by n Pen)
What I Want You To Learn
• How to read a problem statement and translate it into mathematical language
• How to apply our optimization procedure with all details
• How to write a report based on the problem and your solution
Relationship To Course Objectives
• Build critical thinking skills
• Improve mathematical communication skills
• Master calculus techniques
What You Are To Turn In
Please work in groups of 2 or 3 and hand in one report per group. Reports must be very
neat and well organized. They should be written in complete sentences and typed. By
putting your name on the paper you are affirming that all the students in the group
contributed significantly to the effort. The report is expected to be 2 to 3 pages long
including one or two small graphs or pictures if appropriate.
Criteria For Assessment
1. Introductory paragraph(s) and mathematical set-up. The author makes appropriate
use of the given information to explain the purpose of the report and to express
the problem in mathematical terms
2. Mathematical procedures and justifications. The author describes and executes a
clear plan. Mathematical procedures are correctly and thoroughly done. All steps
are explained and justified.
3. Implications and conclusions. All parts of the assignment are addressed.
Important implications and conclusions are clearly stated and explained.
4. Communication and style. Writing style is effective for the stated audience.
Mathematical writing conventions are followed. Grammar, spelling, and wording
choices meet a high standard.
Mathematical Writing Conventions
Study the examples in section 2.5 of our text for good ideas about expository style.
• Notice how the author mixes the necessary mathematical equations with sentences
of logic and explanation.
• Notice that the exposition is almost always in present tense.
• And, the author frequently uses we referring to the reader and author working
together on the steps of solution.
These are all standard conventions in mathematical writing. For more about
mathematical writing, visit http://libguides.lib.muohio.edu/mathwriting
The Problem. The Regional Farm Bureau (RFB) is preparing a brochure that offers
advice about constructing pens for small farm animals, and they want us to be their
consultants. They need us to carefully analyze the following situations and provide a
detailed report. Then they will use our information to help them write the brochure.
In the example they wish to describe, it is assumed that the farmer has 900 feet of fencing
with which to erect a rectangular pen alongside a long, existing fence (so the existing
fence forms one side of the pen). Suppose the pen is to be subdivided into four parts in a
two-by-two arrangement by including interior fences parallel to the outside boundaries.
Then what dimensions make for the largest combined area? What if the farmer
subdivides into nine pens in a three-by-three arrangement? What if the farmer subdivides
into n2 pens in an n-by-n arrangement?
Begin your report with an introduction and general statement of this family of problems.
The best plan is to write a thorough solution of the n-by-n case and then simply apply the
results to the n=2 and n=3 cases. But you may want to focus on the two-by-two case first
in order to more clearly understand what has to be done. Do not break your report into
three, repetitive parts as if you are solving a list of three problems. This should be one
coherent report that deals with all three cases. Be sure to fully justify all conclusions. The
RFB wants to be absolutely sure their brochure will have correct information.
Incorporate into your report a computer-drawn graph of the two functions you are
maximizing in the n=2 and n=3 cases—the functions that give the combined area of the
pen in terms of the overall length of one side of the pen. Use just one set of coordinate
axes with both graphs drawn together. You will need to experiment with the window
dimensions in order to include the important details of the functions. The graphs should
exhibit the domains of the functions and the maximum point of each. Explain in the body
of the report how the graphs differ depending on the value of n. In your explanation,
describe what you would expect to observe if n were 4, 5, 6, … An online graphing utility
can be found at http://www.meta-calculator.com/online/
Your report is directed to the science department of the RFB. You can assume the readers
have studied calculus (years ago, perhaps) and are familiar with the basic terminology.