Superlesson
Project 11-1

Superlesson
Project 11-2

Geometry

Chapter 11, Geometric Inequalities
and Optimization

International and domestic commerce requires that large volumes of different products be transported long distances. Many of these products are shipped by either rail, road, or vessel in large containers shaped like rectangular prisms. These containers come in a few common sizes. Use the Web sites mentioned below to find out how to optimize the shipments.

1. Why is it important for there to be common sizes for these containers?

Part A, Optimizing Areas and Perimeters

2. Go to Zim Service Web site to find the external dimensions of two very common container sizes. As you can see, the two containers have the same width and height, but different lengths. Most containers of this type do have the same width and height, but lengths vary from 10 feet to 56 feet.

a. First, consider only the base of such a container. Given the standard width (in feet) of both of these containers, what length would maximize the ratio of area to perimeter of the rectangular base? Complete the table below to help you arrive at an answer.

Length	Width	Area	Perimeter		Area/Perimeter
	10	8	80	36	2.22
	20	8
	28	8
	40	8
	48	8
	56	8

b. Now, consider the volume of such a container. This time, assume that you must use the standard height given, but can vary the width and length in order to minimize the perimeter of the base. You need to ship 2,176 ft³ of sand and want to use only one container. What are the dimensions of the base?

c. Would a container with the dimensions you found in 2b be practical for transporting substances by rail, road, or vessel?

Part B, Optimizing Volumes and Surface Areas

3. Assume that you want to design a container that has the same length and volume as the 40' container advertised, but uses the least amount of steel. What are the height and width of this container? Approximately how many square feet of steel will you need to construct this container? Explain how you arrived at your answer.