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Calculus 1 Max/Min Writing Project (Landscape Design)
Due date:
Please work in groups of 2 or 3 and hand in one report per group. Reports must be neat and wellorganized.
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 project will be graded both for mathematical quality and for expository quality. Look at the
six examples in section 4.7 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.
The Problem. You are a landscape designer. A client has asked you to design a plan to enclose a
rectangular garden having 10,000 m2 of area. The north and south sides are to be bounded by
wooden fencing, which costs $20/m; the east and west sides are to be bounded by
rhododendrons, which cost $50/m. Find the dimensions of the garden that minimize the total cost
of the fencing and shrubbery. The client is getting estimates from several designers, so you need
to prove that your plan is guaranteed to result in the lowest cost (given those per-meter costs for
installing the fencing and the plants). Accordingly, when you write up your plan include a
careful description of how you find these dimensions and how you verify that these dimensions
actually minimize the cost.
Having finished your plan for the garden to be enclosed by rhododendrons and wooden fencing,
you realize that rectangular gardens with different ”boundary material” on the north and south
than on the east and west have become popular. To simplify the design process, you want to
create a template for these problems. The general situation is as follows. A 10,000 m2 garden is
to be enclosed on the north and south by material A, which costs a dollars per meter, and on the
east and west by material B, which costs b dollars per meter. Find the dimensions that minimize
the total cost of enclosing the garden. (The dimensions will be in terms of the constants a and b.)
As before, in order to prove to your clients that your plan is the best possible, include a careful
description of how you find these dimensions and how you verify that these dimensions actually
minimize the cost.
What does your template tell you about the dimensions of the garden when a = b? When a = 4b?
When a = (1/4)b? Does it agree with the dimensions you found for the rhododendron/wooden
fence-enclosed garden? What happens if the area of the garden is to be something other than
10,000 m2?
The audience for your paper is the client, but assume this client is someone who is familiar with
calculus. Begin with a brief introduction as a reminder of the problem. Incorporate into your
report a computer-drawn graph of the cost function used in the first part of the analysis (with
costs $20/m and $50/m). You will need to experiment with the window dimensions in order to
include the important details of the function. The graph should exhibit the domain of the function
and the minimum point.
Calculus Writing Rubric
1. The purpose, the problem, the given information. This category has to do with
demonstrating a clear understanding of the problem and translating the information of the
problem into useful mathematical terms.
0 to 6 (Emerging) 7 or 8 (Developing) 9 or 10 (Mastering)
Makes little progress in usefully
representing the information or
the problem. Misses the main
ideas and this leads to a trivial or
impossible setup.
Represents the information and
the problem in somewhat
appropriate ways, but with some
missteps or misunderstandings.
Some variables or functions are
unclear or are used
inconsistently.
Demonstrates understanding of
the problem statement or
assignment. Recognizes the
important information and
assumptions. Interprets the given
information accurately. Defines
appropriate variables and
functions and uses them
consistently. Shows insight about
the task.
2. The plan, the procedures. Once the problem statement has been translated into mathematical
terms, now there needs to be a plan for investigating the mathematical situation. It is possible to
set out on a good plan, but make errors in some steps. Both of those are aspects of this category.
0 to 6 (Emerging) 7 or 8 (Developing) 9 or 10 (Mastering)
Does not apply valid procedures
or has not recognized some main
aspects of the assignment.
Applies some appropriate
procedures, but has errors or
oversights.
Formulates and follows a clear
plan. Applies valid and
appropriate procedures. Shows
awareness and understanding of
relevant theories.
3. The explanation, the justification, the implications. In solving a calculus problem, there are
steps of calculation (see #2) and then there are explanations of what is learned from those steps.
0 to 6 (Emerging) 7 or 8 (Developing) 9 or 10 (Mastering)
Provides little or no justification
or explanation of the steps taken.
Justifies most steps, but falls
short at times. Silent about some
interesting implications of the
work.
Explains and justifies all steps.
Supports all assertions. Is
observant about implications and
consequences that follow from
the work. Explains with unusual
precision.
4. The communication, the style. The explanation of each step is part of #3. But how the whole
paper is put together belongs in this category.
0 to 6 (Emerging) 7 or 8 (Developing) 9 or 10 (Mastering)
Unclear organization. Offers no
concluding statement or one that
does not match the problem or
the work. Inconsistencies in
notation. Flaws in logic. Many
problems with writing style.
Graphs or pictures may be
missing or incorrect.
Somewhat clear organization,
but there may be gaps in some
logical connections. Presents
conclusions that are incomplete
or unclear or are based on work
that materially simplified the
task. Some problems with
writing style. Graphs or pictures
are included, if applicable.
Communicates effectively and
with appropriate style. The
introduction, body, and
conclusion hang together.
Presents a clear, complete and
logical exposition. Goes beyond
the basic requirements of the
assignment. Graphs or pictures,
if applicable, are right on target.