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Chapter 11 Assignments 167

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Whirlygigs for Sale!Rotating Two-Dimensional Figures through Space

The ChocoWorld Candy Company is going to enter a candy competition in which they will make a structure entirely out of chocolate. They are going to build a fairytale castle using several different molds, and they need to make the molds using a drill bit that will create the shape they are striving for.

1. The castle will need several turrets, which are made by pouring chocolate into a mold that willform a cone.

a. Which of the figures shown is a cone?

Fig. 1 Fig. 2 Fig. 3 Fig. 4

Figure 3 is a cone.

b. Which of the drill bits shown will form a cone after being rotated in a plastic molding compound?

A B C D

Drill bit A will form a cone after being rotated in a plastic molding compound.

Lesson 11.1 Assignment

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168 Chapter 11 Assignments

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c. What is the shape on the drill bit that forms the cone in the mold as it is being rotated?

The shape on the drill bit that forms the cone is a triangle.

d. If the triangle on the drill bit is 2 inches wide and 1 inch tall, what will the dimensions of the cone be that is formed by the rotation of the bit?

The radius of the cone will be half of the base of the triangle, or 1 inch. The height of the cone will be the same as the height of the triangle, 1 inch.

2. The castle that the company is making is going to have long circular columns in the front.

a. What type of solid mold is needed to create circular columns?

A cylindrical mold is needed to create circular columns.

b. Which of the drill bits shown will form the mold needed after being rotated in the plasticmolding compound?

A

B C D

Drill bit C will form a cylinder that models a long circular column.

c. What is the shape of the drill bit that will create the column mold?

The shape of the drill bit is a rectangle.

d. If the width of the shape on the end of the drill bit being rotated is 3 inches, what is the radius of the base of the cylinder going to be?

The radius of the base of the cylinder will be 1.5 inches.

Lesson 11.1 Assignment page 2

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3. To complete the castle, the company is going to create small cannonballs for the cannon that will be situated on the roof of the castle.

a. What type of solid mold is needed to make cannonballs?

A spherical mold is needed to create cannonballs.

b. Which of the drill bits shown will form the mold needed if the cannonballs will have a radius of 0.25 inch?

A B C D

3 in. 0.25 in.

0.25 in.1 in.

Drill bit B will form a spherical mold with a radius of 0.25 inch.

c. What is the shape of the drill bit that will create the cannonball mold?

The shape of the drill bit is a circle.


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Cakes and PancakesTranslating and Stacking Two-Dimensional Figures

Theodore is starting a new company that will manufacture food storage containers. He asks his engineers to design several different containers based on which type of containers will sell the best.

1. The end of one container is a rectangle

a. Translate the rectangle in a diagonal direction to create a second rectangle.

See diagram.

b. Use dashed line segments to connect each pair of corresponding vertices in the rectangles.

See diagram.

c. What do you notice about the relationship among the line segments in your drawing?

The line segments appear to be parallel and congruent to each other.

d. What is the name of the solid formed by this translation?

The solid formed by translating a rectangle is a rectangular prism.


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2. The base of one container is a triangle.

a. Translate the triangle in a diagonal direction to create a second triangle.

See diagram.

b. Use dashed line segments to connect each pair of corresponding vertices in the triangles.

See diagram.

c. What do you notice about the relationship among the line segments in your drawing?



The solid formed by translating a triangle is a triangular prism.

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3. Theodore gets a contract from a restaurant to make containers for their soup. The bottom of the container is a disc. An oval is drawn to represent what the base of the container might look like.

a. Translate the oval in a diagonal direction to create a second oval.

See diagram.

b. Use dashed line segments to connect the tops and the bottoms of the ovals.

See diagram.

c. What do you notice about the relationship between the line segments in your drawing?



The solid formed by translating a disc is a cylinder.

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The Harrington Heights Middle School is getting ready for its spring musical. The students in the art classes are using donated cardboard boxes to make the props.

4. One of the props for the show needs to be a suitcase. The directors want its weight to be light because it is used in one scene to playfully hit another actor. The art students decide to make the suitcase shape by stacking rectangles of the same shape and size on top of each other. Each cardboard cutout is 48 inches long by 30 inches wide.

a. The students are going to stack 4 rectangular cutouts on top of each other. What is the name of the solid formed by this stack of rectangles?

The solid formed by stacking the rectangular cutouts is a rectangular prism.

b. If the thickness of each cardboard rectangle is 0.25 inch, what is the thickness of the suitcase?

The thickness of the suitcase will be 1 inch.

0.25 3 4 5 1

c. Relate the dimensions of the suitcase formed to the dimensions of the rectangles.

The length of the suitcase is the same as the length of the rectangles, or 48 inches. The width of the suitcase is the same as the width of the rectangles, or 30 inches.

d. The directors of the show have decided that they would like the suitcase to have a greater thickness than 1 inch. They would like it to have a thickness of 3 inches. How many rectangles do the students need to stack?

The students know that 4 rectangles make a suitcase that is 1 inch high. Therefore, 12 rectangles will make a suitcase that is 3 inches high.


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5. The art students need to work on making a stop sign for the show using stacks of figures that have similar shapes and sizes.

a. What type of shape will they need to cut out from the cardboard for their stack?

The students will need to cut out octagons.

b. What is the name of the solid that will be formed by this stack?

The solid formed by stacking the octagonal cutouts is an octagonal prism.

c. The cardboard the students use for the stop sign is 0.125 inch thick. If the directors want the stop sign to be 1 inch thick, how many stop signs will the students need to stack?

The students will need to stack 8 octagonal cutouts because 8 3 0.125 5 1.

6. One of the scenes of the musical involves a scene on city streets. The director would like to have several traffic cones set up for the scene.

a. The students need to build up each cone using stacks of shapes. What type of shapes will they use? What should the size of the shapes be?

The students will use discs stacked on top of each other to make a cone. Each additional disc that gets stacked should have a slightly smaller radius than the previous disc.

b. What is the relationship among the discs getting stacked?

The discs are similar discs. They are the same shape but not the same size.

c. The cardboard they are using for the cone is 0.5 inch thick. The directors want the traffic cones to be 14 inches tall. How many discs will the students need to stack for each cone?

The students will need to use 28 discs because 0.5 3 28 5 14.

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Cavalieri’s PrinciplesApplication of Cavalieri’s Principles

l

h

1. Divide the figure shown into approximately 10 rectangles. What is the length, the height, and the area of each rectangle?

The length of each rectangle is ¯ ___ 10

, the height of each rectangle is h, and the area of each

rectangle is ̄ h ___ 10

.

2. What is the approximate area of the irregularly shaped figure?

The approximate area of the irregularly shaped figure is ̄ h ___ 10

? 10 5 ¯h.

3. If this irregularly shaped figure were divided into 1000 congruent rectangles, what would be the approximate area of the figure?

The approximate area of the irregularly shaped figure would be ¯h _____ 1000

? 1000 5 ¯h.

4. If this irregularly shaped figure were divided into n congruent rectangles, what would be the approximate area of the figure?

The approximate area of the irregularly shaped figure would be ̄ h ___ n ? n 5 ¯h.


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The Leaning Tower of Pisa in Italy is about 180 feet tall from the top of the tower vertically to the ground. It has a diameter of approximately 51 feet.

5. Determine the approximate volume of the tower. Explain your reasoning.

The tower is shaped like an oblique cylinder. Cavalieri’s principle tells me that an oblique cylinder has the same volume as a right cylinder with the same height and base area.

The volume formula is p r 2 h.

p (25.5 ) 2 (180) ¯ 367,707.71

The Leaning Tower of Pisa has a volume of approximately 367,707.71 cubic feet.


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Spin to WinVolume of Cones and Pyramids

1. Joel owns a frozen yogurt and fruit smoothie shop. He just placed an order for three different sizes of cones. He needs to determine how much to charge for each cone and decides that knowing the volume of each might help him make his decision.

1.875"

Cone 1

4.5"

1.625"

Cone 2

7"

Cone 3

6"

2 12"

a. Which cone do you think has the greatest volume? Explain your reasoning.

Answers will vary.

Example Response: I think cone 3 has the greatest volume because it looks bigger than the other cones.

b. Identify the radius, the diameter, and he height of cone 1. How did you determine the radius of the cone?

The radius is 0.9375 inch, the diameter is 1.875 inches, and the height is 4.5 inches.

I divided the diameter in half to get the radius.


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c. Calculate the volume of cone 1. Show your work. Round your answer to the nearest hundredth.

V 5 1 __ 3 r 2 h

5 ( 1 __ 3 ) (3.14)(0.9375 ) 2 (4.5)

¯ 4.14 cubic inches

The volume of cone 1 is approximately 4.14 cubic inches.

d. Identify the radius, the diameter, and the height of cone 2. How did you determine the diameter of the cone?

The radius of cone 2 is 1.625 inches, the diameter is 3.25 inches, and the height is 7 inches.

I doubled the radius to get the diameter.

e. Calculate the volume of Cone 2. Show your work. Round to the nearest hundredth.

V 5 1 __ 3 r 2 h

5 ( 1 __ 3 ) (3.14)(1.625 ) 2 (7)


The volume of cone 2 is approximately 19.35 cubic inches.

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f. Identify the radius, the diameter, and the height of cone 3. How did you determine the radius of the cone?

The radius of cone 3 is 1.25 inches, the diameter is 2.5 inches, and the height is 6 inches.

I divided the diameter in half to get the radius.

g. Calculate the volume of Cone 3. Show your work. Round to the nearest hundredth.

V 5 1 __ 3 r 2 h

5 ( 1 __ 3 ) (3.14)(1.25 ) 2 (6)


The volume of Cone 3 is approximately 9.81 cubic inches.

h. Determine which size cone Joel should charge the most for and the least for. Explain your reasoning.

Joel should charge the most for cone 2 because it has the greatest volume.

Joel should charge the least for cone 1 because it has the least volume.

i. Compare the volumes of all three cones.

The volume of cone 2 is almost five times the volume of cone 1 and almost two times the volume of cone 3.


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2. Pyramid tents were popular for a time during the 19th century. Although their popularity declined during the 20th century, they have recently begun to regain popularity again. The design is ideal for shaping canvas, and it only requires one pole and some stakes to secure it. Joe wants to make a right square pyramid tent and is considering two different sizes. He will either make one with a base that is 10 feet by 10 feet and has a height of 12 feet, or he will make one with a base that is 12 feet by 12 feet and has a height of 8 feet.

a. Sketch the two pyramid designs Joe is considering and label them with the given measurements.

Sketches will vary.

12 ft

10 ft

8 ft

12 ft

b. How can you determine which pyramid tent will have the most interior space?

I can calculate the volume of each pyramid tent to determine how much interior space each one will have.

c. Calculate the volume of each proposed pyramid tent. Show your work.

Tent With 12-Foot Height: Tent With 8-Foot Height:

V 5 1 __ 3 b 2 h V 5 1 __

3 b 2 h

V 5 1 __ 3 (10 ) 2 (12) V 5 1 __

3 (12 ) 2 (8)

V 5 400 cubic feet V 5 384 cubic feet

d. Which tent would you recommend Joe make? Explain your reasoning.

Answers will vary.

Example Response: I would recommend the tent with the 12-foot height because it has more interior room.


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Spheres à la ArchimedesVolume of a Sphere

Calculate the volume of each sphere. Use 3.14 for p and round to the nearest tenth, if necessary.

1.

21 mm

2.

45 ft

Volume 5 4 __ 3 p r 3

5 4 __ 3 p (10. 5 3 )

5 1543.5p

¯ 4846.6 square millimeters

Volume 5 4 __ 3 p r 3

5 4 __ 3 p (4 5 3 )

5 121,500p

¯ 381,510 square feet

3. A can holds 3 tennis balls as shown in the figure. The radius of each tennis ball is 3 centimeters.

a. What is the volume of a single tennis ball?

Volume 5 4 __ 3 p r 3

5 4 __ 3 p ( 3 3 )

¯ 4 __ 3 (3.14) (27)

¯ 113.04 cubic centimeters

b. What is the total volume all 3 tennis balls take up?

Total volume ¯ 3(113.04)


3 cm

3 cm

3 cm


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c. Can you determine the height of the can? Explain your reasoning.

Yes. The height of the can is 18 centimeters. The height of the can is three times the diameter of one tennis ball.

height of can 5 3 3 diameter of one tennis ball

5 3 3 6

5 18

d. What is the volume of the can? Use 3.14 for p.

Volume 5 p r 2 h

¯ (3.14)( 3 2 )(18)


e. What is the volume of the can not taken up by the tennis balls?

The volume of the can that is not taken up by the tennis balls is approximately 169.56 cubic centimeters.

508.68 2 339.12 ¯ 169.56

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Turn Up the . . .Using Volume Formulas

1. The Luxor Hotel in Las Vegas is a replica of the Pyramid of Khafre at Giza, one of the seven wonders of the world. The Luxor’s base is a square with a side length of 646 feet, and it is 350 feet tall.

a. What is the volume of the Luxor Hotel?

V 5 1 __ 3 Bh

5 1 __ 3 (64 6 2 )(350)

5 48,686,866 2 __ 3

The volume of the Luxor Hotel is 48,686,866 2 __ 3 cubic feet.

b. The Pyramid of Khafre has a volume of 2,226,450 cubic meters. Its base is a square with a side length of 215 meters. What is the height of the Pyramid of Khafre?

V 5 1 __ 3 Bh

2,226,450 5 1 __ 3 (21 5 2 )h

144.5 ¯ h

The height of the Pyramid of Khafre is approximately 144.5 meters.


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2. A store sells square pyramid-shaped scented candles. The dimensions of two of the candles are shown.

6 cm6 cm

16 cm

Candle A

8 cm 8 cm

9 cm

Candle B

a. Calculate the volume of each candle.

The volume of candle A is 192 cubic centimeters. The volume of candle B is also 192 cubic centimeters.

V 5 1 __ 3 Bh V 5 1 __

3 Bh

5 1 __ 3 ( 6 2 )(16) 5 1 __

3 ( 8 2 )(9)

5 192 5 192

b. Both candles are made of wax. Which candle contains more wax? Explain.

I know both candles contain the same amount of wax because their volumes are equal.

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3. Your municipality is replacing the storage tanks in the community. Which plan provides the greater total capacity?

Plan 1: Install one cylindrical tank that is 150 feet tall and has a radius of 50 feet.

Plan 2: Install two cylindrical tanks that are 75 feet tall. One cylindrical tank has a radius of 30 feet, and one tank has a radius of 25 feet.

Use 3.14 for p and round your answers to the nearest tenth if necessary.

Plan 1 provides a greater total capacity.

Volume for plan 1:

V 5 p (50 ) 2 (150)

¯ (3.14)(2500)(150)

¯ 1,177,500 cubic feet

Volume for plan 2:

V 5 p (3 0 2 )(75) 1 p (2 5 2 )(75)

¯ (3.14)(900)(75) 1 (3.14)(625)(75)

¯ 211,950 1 147,187.5

¯ 359,137.5 cubic feet

4. A traffic cone has a radius of 9 inches and a height of 30 inches. What is the volume of thistraffic cone?

The volume of the traffic cone is approximately 2543.4 cubic inches.

V 5 1 __ 3 p r 2 h

5 1 __ 3 p ( 9 2 )(30)

5 810p

¯ 2543.4 in . 3

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5. A funnel that is used to change the oil in a car is in the shape of a cone. The base of the funnel has a circumference of 60 centimeters. The height of the funnel is 25 centimeters. How much oil will this funnel hold?

The volume of the funnel is approximately 2388.5 cubic centimeters.

C 5 2p r V 5 1 __ 3 p r 2 h

60 5 2p r 5 1 __ 3 p ( 60 ___

2p ) 2 (25)

60 ___ 2p

5 r ¯ 2388.5 cubic centimeters

6. Today’s deal at the ice cream shop is a mini cone with one scoop of ice cream.

a. A mini ice cream cone has a diameter of 3.5 centimeters and a height of 6 centimeters. How much ice cream fits in the cone?

The volume of the ice cream cone is approximately 19.2 cubic centimeters.

V 5 1 __ 3 p r 2 h

5 1 __ 3 p (1.7 5 2 )(6)

5 6.125p


b. One scoop of ice cream has the same diameter as the cone, 3.5 centimeters. What’s the volume of 1 scoop of ice cream?

The volume of one scoop of ice cream is approximately 22.45 cubic centimeters.

V 5 4 __ 3 p r 3

5 4 __ 3 p (1.7 5) 3

5 343 ____ 48

p

¯ 22.45


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Tree RingsCross Sections

Describe the shape of each cross section.

1. 2.

The cross section is a point. The cross section is a square.

3. 4.

The cross section is a triangle. The cross section is a line.

5. 6.

The cross section is a hexagon. The cross section is a triangle.


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7. Sketch two cross sections of a pentagonal prism—one cross section that is parallel to the base and another cross section that is perpendicular to the base.

A cross section parallel to the base A cross section perpendicular to the base

is a pentagon. is a rectangle.

8. Sketch two cross sections of a cone—one cross section that is parallel to the base and another cross section that is perpendicular to the base.

A cross section parallel to the base is a circle. A cross section perpendicular to the base that passes through the apex of the cone is a triangle.

9. A solid’s cross section parallel to the base is an octagon. A cross section of the solid perpendicular to the base is a triangle. Identify the solid.

The solid is an octagonal pyramid.

10. A solid’s cross section parallel to the base is a triangle. A cross section of the solid perpendicular to the base is a rectangle. Identify the solid.

The solid is a triangular prism.

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Two Dimensions Meet Three DimensionsDiagonals in Three Dimensions

1. What is the length of a three-dimensional diagonal of the rectangular prism?

4 cm

6 cm11 cm

The length of a three-dimensional diagonal of the rectangular prism is about 13.2 centimeters.

d 5 √_____________

112 1 62 1 42

5 √______________

121 1 36 1 16

5 √____

173

¯ 13.2

2. What is the length of a three-dimensional diagonal of the rectangular prism?

12 in.

15 in.

18 in.

The length of a three-dimensional diagonal of the rectangular prism is about 18.6 inches.

d 5 √__________________

1 __ 2

(182 1 152 1 122 )

5 √___________________

1 __ 2 (324 1 225 1 144 )

5 √______

346.5

¯ 18.6


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3. A rectangular box has a length of 6 feet and a width of 2 feet. The length of a three-dimensional diagonal of the box is 7 feet. What is the height of the box?

The height of the box is 3 feet.

d 2 5 2 1 w2 1 h2

72 5 62 1 22 1 h2

49 5 36 1 4 1 h2

9 5 h2

3 5 h

4. The length of the diagonal across the front of a rectangular box is 20 inches, and the length of the diagonal across the side of the box is 15 inches. The length of a three-dimensional diagonal of the box is 23 inches. What is the length of a three-dimensional diagonal of the box?

The length of the diagonal across the top of the box is about 20.81 inches.

d 2 5 1 __ 2 (d1

2 1 d22 1 d3

2)

232 5 1 __ 2 (202 1 152 1 d3

2)

529 5 1 __ 2 (400 1 225 1 d3

2)

1058 5 625 1 d32

433 5 d32

20.81 ¯ d3

5. Pablo is packing for a business trip. He is almost finished packing when he realizes that he forgot to pack his umbrella. Before Pablo takes the time to repack his suitcase, he wants to know if the umbrella will fit in the suitcase. His suitcase is in the shape of a rectangular prism and has a length of 2 feet, a width of 1.5 feet, and a height of 0.75 foot. The umbrella is 30 inches long. Will the umbrella fit in Pablo’s suitcase? Explain your reasoning.

The umbrella will fit in the suitcase because its length is 2.5 feet and the length of the three-dimensional diagonal of the suitcase is approximately 2.6 feet.

Length of Three-Dimensional Diagonal of Suitcase:

d 5 √________________

22 1 1.52 1 0.752

5 √__________________

4 1 2.25 1 0.5625

5 √_______

6.8125

¯ 2.6 feet

Length of Umbrella:

30 ___ 12

5 2.5 feet

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