Superlesson
Project 11-1

Superlesson
Project 11-2

Geometry

Chapter 11 Answers Geometric Inequalities and Optimization

Orienteers use a detailed map and compass to navigate a cross-country course. People enjoy orienteering as both a nice way to get some exercise in a natural setting as well as a competitive sport. Go to the Orienteering Web site to read about orienteering. In this activity, you will analyze a course on an orienteering map provided by the University of Bristol Orienteering Club in England.

Part B, Inequalities in a Triangle

1. Go to the Ashton Court Web site to see a map of Ashton Court in Bristol. You can use angle measurements to approximate relative distances in orienteering. In this activity, you will perform such an analysis for the section of the course between control sites #8 and #11.

1. Refer to the diagram below showing the relative positions of control sites #8 through #11.

a. Find the approximate angle measures for each of the six angles shown below using your map. Write these measurements on your diagram.

Answer:

b. As you go from control site #8 to control site #11, you will have walked three separate distances. Which of these distances is longest and which is shortest?
[#8 to #9 is shortest and #10 to #11 is longest.]

Part C, The Triangle Inequality Theorem

2. Return to the Ashton Court Web site to answer the following questions.

a. You're doing the Ashton Court course for fun one day and decide to estimate the distances between each of the control sites. You estimate that the distance from site #2 to site #3 is 270 meters and the distance between site #3 and site #4 is 250 meters. When you get to site #4, you realize that you left your rain jacket back at control site #2. Mathematically, what is the minimum distance you will have to travel to get back to site #2? What is the maximum distance?
[minimum = 20 meters; maximum = 520 meters.]

b. Approximate the true distance you will have to travel based on the map.
[about 300 meters]

Top

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?
[All of the machinery, platforms and storage areas used to load and ship containers are used most flexibly if they can accommodate many different types of containers. This requires that containers have fairly standard dimensions. ]

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

[A length of 56' would maximize the area to perimeter ratio.]

Length	Width	Area	Perimeter		Area/Perimeter
	10	8	80	36	2.22
	20	8	160	56	2.86
	28	8	224	72	3.11
	40	8	320	96	3.33
	48	8	384	112	3.43
	56	8	448	128	3.50

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?
[16' x 16']

c. Would a container with the dimensions you found in 2b be practical for transporting substances by rail, road, or vessel?
[Answers may vary. However, this would probably only be a practical container for water transportation because it would be too wide for roads or railways. If a cargo ship was designed to fit two standard conatiners side-by-side, it would fit one of the containers with the dimensions above.]

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.
[height = width ≈ 8.25 ft. You'll need a little bit more than 1456 ft² of steel to construct this container.