Geometry
Chapter 9 Answers
Surface Area and Volume
Once packages are designed, they may be painted or labelled for decoration or identification. It is necessary to find the areas of the sides and faces of these containers first.
Part A, Surface Area of Prisms
1. Go to the Jos Steeman
Alkmaar BV Web site to see dimensions for three different sized
cargo containers. Each of these containers has six sides made of steel.
Find the surface area of the 20 Foot Dry Cargo Container in order to estimate
the amount of steel necessary to manufacture this container. Use square
feet for the surface area.
[796 sq. ft.]
Part C, Surface Area of Cylinders and Cones
2. It is important to be able to calculate the surface area of containers in order to manufacture labels for these containers. Go to the Duvivier Web site to see an advertisement for a can labeller. You will use the technical data for Model D210 in this exercise.
a. Use the technical data at the bottom of this web site to find the minimum circumference of a container which labeller model D210 can create a label for.
[157 mm]b. Why is the minimum length of the label actually longer than this?
[To allow for overlap.]c. Find the minimum lateral surface area of a container which labeller model D210 can create a label for.
[6751 sq. mm]d. What is the minimum area of a label model D210 can create?
[5270 sq. mm]e. Why is the minimum area of the label actually less than the minimum lateral surface area of the can?
[The lable does not go from the bottom edge to the top edge of the can. There is a border above and below the label.]f. Why is the maximum container height 8 mm more than the maximum label width?
[To allow for ends on cans and lids on jars.]
3. Some containers are made of a combination of three-dimensional shapes.
Go to Conical
bottom Web site to see dimensions of tanks which are shaped like
a cylinder placed on top of a cone. Look at the dimensions for model CBO-45.
Suppose that the cone and the cylinder each have a height of 21". Find
the lateral surface area of this container (excluding the top).
[2270.7 sq. in]
Many naturally occurring objects have shapes which may be approximated by simple three-dimensional figures such as prisms, cylinders, cones and spheres. Because we know how to find the volumes of such figures, this gives us an easy method to estimate the volumes of many objects in nature.
Part A, Volume of Prisms
1. Go to Mother Nature's tile floor Web site to see a picture of the top of Devil's Postpile.
a.Read the description of this amazing natural formation. It is composed of many vertical shafts, some shaped like regular hexagonal prisms. What is the approximate diameter of each of these shafts?
[1.5 ft]b. To see a front view of Devil's Postpile, go to this Web site.
c. Find the approximate height of Devil's Postpile in the first paragraph of the Devil's Postpile Reds Meadow Web site.
[60 ft]d. Assuming that the "diameter" in 1a is drawn as in the figure below, find the approximate volume of one of the vertical columns with a hexagonal base in Devil's Postpile.
[87.7 cubic ft]
Part C, Volume of Cylinders and Cones
2. Use the Internet to answer the following questions.
a. The trunk of a tree can be approximated by a cylinder. Go to the Coast Redwood Primer Web site to see some information about the coast redwoods. What is the approximate radius of the base of "Tall Tree?"
[About 7 ft]b. Find the approximate volume for the trunk of "Tall Tree," the tallest coast redwood anywhere.
[56618 cubic ft]c. Do you believe your approximation is greater or less than the actual volume of the trunk of "Tall Tree?"
[Probably greater because the height of the tree most likely takes into account branches which rise above the top of the trunk, and the base is most likely the widest part of the trunk.]
3. Volcanoes are approximately conical in shape. Go to the Mount Shasta Volcano Web site to see information from various sources about Mount Shasta, a volcano in northern California. For this project, use the information given by Wood and Kienle (1990), the fourth source from the top.
a. What is the height from the base of Mount Shasta to its top in kilometers? What is its approximate volume?
[height = 3.5 km; volume = 350 cubic km]b. Use the information in 3a to find the approximate circumference of the base of Mount Shasta.
[About 61 km]c. Why is it important to know the volume of a volcano?
[Volume gives an approximation of the amount of material which could be expelled in an eruption.]
Part D, Surface Area and Volume of Spheres
4. Spheres are good approximations for the shapes of both the Earth and
the Moon. Go to the Earth
Web site to get dimensions for both the Earth and the Moon. How many times
bigger is the Earth's volume than the Moon's?
[About 49 times bigger.]
Collies look like bigger versions of Shetland Sheepdogs, otherwise known as Shelties. Below, you will use measurements for the bodies of each of these canines to explore whether these two animals truly have mathematically similar shapes.
Part A, Surface Area of Similar Solids
1. You will refer to two Web sites in this activity. The Web site which gives the breed standards for Collies and the Shetland Sheepdogs Web site.
a. For now, let's assume that the Collie and the Sheltie are mathematically similar. Use the average of the minimum and maximum heights at shoulder to find the similarity ratio between a female Collie and a Sheltie. Then, find the similarity ratio between a male Collie and a Sheltie.
[female Collie : Sheltie = 23 : 14.5; male Collie : Sheltie = 25 : 14.5]b. Use these similarity ratios to find the ratio of the surface areas between a female Collie and a Sheltie. Then, find the similarity ratio between the surface areas of a male Collie and a Sheltie.
[female Collie : Sheltie = 529 : 210.25; male Collie : Sheltie = 625 : 210.25]c. What does the surface area of an animal actually measure?
[The amount of skin the animal has.]
Part B, Volume of Similar Solids
3. You will now investigate whether the Sheltie is actually mathematically similar to either the female or male Collie. You will use the ratio of their weights to determine this. For this exercise, you can assume that the weight ratio and volume ratio are numerically equivalent.
a. Assuming the female Collie is mathematically similar to the Sheltie, what should the ratios of their weights be?
[12167 : 3048.625]b. Given the weight of the Sheltie, about what should a female Collie weigh if they are mathematically similar?
[Close to 80 pounds.]c. Are a female Collie and a Sheltie mathematically similar?
[Only an extraordinarily heavy female Collie could be similar to the Sheltie.]d. Are a male Collie and a Sheltie mathematically similar? Why or why not?
[A male Collie would have to weigh about 102.5 pounds to be mathematically similar to a Sheltie. Once again, this would be an extremely heavy male Collie.]