Question

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.

Answer 1