Name Class Date 11-1 - Pequannock Township High Schoolhs.pequannock.org/ourpages/auto/2014/8/26/59171599/ch11... · 2015-04-14 · Name Class Date 11-1 Think About a Plan Space Figures

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Name Class Date

11-1 Additional Vocabulary SupportSpace Figures and Cross Sections

Concept List

cross section edge Euler’s Formula

face horizontal plane net

polyhedra vertex vertical plane

Choose the concept from the list above that best represents the item in each box.

1. 2. F 1 V 5 E 1 2 3.

4. BF 5. 6. A

7. quadrilateral ABCD 8. plane J 9. plane K

A B

C

H

D

E F

G A B

C

H

D

E F

G

A B

CD

E F

H GJ

K

polyhedra

edge

face

horizontal plane vertical plane

cross section

vertex

Euler’s Formula

net


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11-1 Think About a Plan Space Figures and Cross Sections

Visualization Draw and describe a cross section formed by a plane intersecting the cube as follows.

Th e plane is tilted and intersects the front and back faces of the cube perpendicular to the left and right faces.

Understanding the Problem

1. What is a cube? Draw a typical view of a cube and describe a cube.

2. What is a plane? Draw a typical view of a plane and describe a plane.

3. What is a cross section? Draw a plane passing through your cube that is parallel to the top and bottom faces of your cube. Explain why the cross section is a square.

Planning the Solution

4. How can you use your understanding of cubes, planes, and cross sections to draw a plane that is tilted and intersects the front and back faces of the cube perpendicular to the left and right faces?

5. You showed that if a plane is not tilted, the cross section is a square. How can you use this knowledge to predict what the cross section will look like if the plane is tilted?

Getting an Answer

6. Prepare to draw a plane that is tilted and intersects the front and back faces of the cube perpendicular to the left and right faces by studying your drawing of a plane parallel to the top and bottom faces. Imagine the thin slice created when this tilted plane passes through the cube. Describe this cross section.

a rectangular prism with six square faces, all congruent

a two-dimensional surface unbounded in all directions

A cross section is an

intersection of a solid and a plane. It is a very thin slice of the

solid. Because the plane is parallel to two opposing faces, the cross section is congruent

to the faces. So, the cross section is a square congruent to the faces.

Look at the drawing of a parallel plane to understand how a tilted plane is different.

Contrast the cross section you get when the plane is parallel to the cross

section you will get when the plane is tilted.

The cross section is a rectangle. Two sides are the same length as the edges of the cube,

and two sides are longer than the edges.

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11-1 Practice Form G

Space Figures and Cross Sections

For each polyhedron, how many vertices, edges, and faces are there? List them.

1.

2.

For each polyhedron, use Euler’s Formula to fi nd the missing number.

3. Faces: u Edges: 12 Vertices: 8

4. Faces: 9 Edges: u Vertices: 14

5. Faces: 10 Edges: 18 Vertices: u 6. Faces: u Edges: 24 Vertices: 16


Verify Euler’s Formula for each polyhedron. Th en draw a net for the fi gure and verify Euler’s Formula for the two-dimensional fi gure.

8. 9.

Describe each cross section.

10. 11. 12.

P

EA

B C

D

A BCD

GH

FE

6 vertices: A, B, C, D, E, P; 10 edges: AP, BP, CP, DP, EP, AB, BC, CD, DE, EA; 6 faces: kAPB, kBCP, kCDP, kDEP, kEAP, and pentagon ABCDE

8 vertices: A, B, C, D, E, F, G, H; 12 edges: AB, BC, CD, DA, AE, BF, CG, DH, EF, FG, EH, GH; 6 faces: quadrilaterals ABCD, ABFE, CBFG, DCGH, ADHE, EFGH

6

21

10

10

12

a hexagona circle

polyhedron: 7 faces, 15 edges, 10 vertices; F 1 V 5 E 1 2, 7 1 10 5 15 1 2, 17 5 17; net: 7 faces, 24 edges, 18 vertices; F 1 V 5 E 1 1, 7 1 18 5 24 1 1, 25 5 25

polyhedron: 6 faces, 12 edges, 8 vertices; F 1 V 5 E 1 2, 6 1 8 5 12 1 2, 14 5 14; net: 6 faces, 19 edges, 14 vertices; F 1 V 5 E 1 1, 6 1 14 5 19 1 1, 20 5 20

a parallelogram


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13. Open-Ended Sketch a polyhedron with more than four faces whose faces are all triangles. Label the lengths of its edges. Use graph paper to draw a net of the polyhedron.

Use Euler’s Formula to fi nd the number of vertices in each polyhedron.

14. 6 faces that are all parallelograms

15. 2 faces that are heptagons, 7 rectangular faces

16. 6 triangular faces

Reasoning Can you fi nd a cross section of a square pyramid that forms the fi gure? Draw the cross section if the cross section exists. If not, explain.

17. square 18. isosceles triangle 19. rectangle that is not a square

20. equilateral triangle 21. scalene triangle 22. trapezoid

23. What is the cross section formed by a plane containing a vertical line of symmetry for the fi gure at the right?

24. What is the cross section formed by a plane that is parallel to the base of the fi gure at the right?

25. Reasoning Can a polyhedron have 19 faces, 34 edges, and 18 vertices? Explain.

26. Reasoning Is a cone a polyhedron? Explain.

27. Visualization What is the cross section formed by a plane that intersects the front, right, top, and bottom faces of a cube?

11-1 Practice (continued) Form G


Answers may vary. Sample drawing is at the right.

No; any plane that intersects the four faces has either a trapezoid, a square, or no parallel sides in its cross section.

triangle

octagon

No; by Euler’s Formula, there should be 35 edges, 18 faces, or 17 vertices.

No; all of the faces of a polyhedron are polygons. A cone has no faces that are polygons.

a rectangle or a trapezoid

8

14

5

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11-1 Practice Form K


For each polyhedron, how many vertices, edges, and faces are there? List them.

1. Vertices:

Edges: AB, AF , AD, BE, BC, CD, CH , HE, HG, GF , EF , DG

Faces:

2. Vertices:

Edges: XY , XW , XZ, YV , YZ, VW , VZ, WZ

Faces:

For each polyhedron, use Euler’s Formula to fi nd the missing number.

3. Faces: u Edges: 8 Vertices: 5

To start, use Euler’s formula, then identify F 1 V 5 E 1 2the variables and any given values.


5. Faces: 4 Edges: 6 Vertices: uVerify Euler’s Formula for each polyhedron. Th en draw a net for the fi gure and verify Euler’s Formula for the two-dimensional fi gure.

6. 7.

Use Euler’s Formula to fi nd the number of vertices in each polyhedron.

8. 6 faces that are all squares

9. 1 face that is a hexagon, 6 triangular faces

10. 2 faces that are pentagons, 5 rectangular faces

11. Reasoning Can a polyhedron have 20 faces, 30 edges, and 13 vertices? Explain.

12. Reasoning Is a cylinder a polyhedron? Explain.

A

B CD

E

FG

H

Z

W

V

X

Y

A, B, C, D, E, F, G, H; 8

VWZ, XYZ, WXYV, XWZ, YVZ; 5

No; all of the faces of a polyhedron are polygons. A cylinder has faces that are circles, and circles are not polygons.

8 faces, 18 edges, 12 vertices; 8 1 12 5 18 1 28 faces, 29 edges, 22 vertices; 8 1 22 5 29 1 1

4 faces, 6 edges, 4 vertices; 4 1 4 5 6 1 2 4 faces, 9 edges, 6 vertices; 4 1 6 5 9 1 1

12

5

ABCD, ABEF, ADGF, DCHG, BCHE, EFGH; 6

12

4

V, W, X, Y, Z; 5

8

7

10

8

No; by Euler’s Formula, there should be 31 edges, 19 faces, or 12 vertices.


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Describe each cross section.

13. To start, visualize the plane’s intersection with the solid.

14. 15.

Reasoning Can you fi nd a cross section of a square pyramid that forms the fi gure? Draw the cross section if the cross section exists. If not, explain.

16. isosceles triangle 17. trapezoid

18. scalene triangle 19. square

20. What is the cross section formed by a plane containing a vertical line of symmetry for the fi gure at the right?

21. What is the cross section formed by a plane that is parallel to the base of the fi gure at the right?

11-1 Practice (continued) Form K


a rectangle

rectangle

a triangle

hexagon

an oval (ellipse)


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11-1 Standardized Test Prep Space Figures and Cross Sections

Multiple Choice

For Exercises 1–5, choose the correct letter.

1. A polyhedron has 6 vertices and 9 edges. How many faces does it have?

3 5 7 9

2. A polyhedron has 25 faces and 36 edges. How many vertices does it have?

11 12 13 14

3. Which of the following shows a net for a solid that has 8 faces, 12 vertices, and 18 edges?

4. What is the cross section formed by a plane that contains a vertical line of symmetry for a tetrahedron?

triangle square rectangle pentagon

5. What is the cross section formed by a plane that intersects three faces of a cube?

triangle square rectangle pentagon

Short Response

6. How many edges and vertices are there for an octahedron, a polyhedron with eight congruent triangular faces?

B

F

A

D

H

[2] 12 edges AND 6 vertices [1] 12 edges OR 6 vertices [0] no correct response given


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11-1 Enrichment Space Figures and Cross Sections

Archimedean SolidsAn Archimedean solid is a polyhedron that has a similar arrangement of polygons about each vertex. Th e faces of an Archimedean solid are made up of two or more types of regular polygons.

In total, there are 13 Archimedean solids. One example is the icosidodecahedron, shown in the drawings at the right.

1. What are the regular polygons that make up the faces of an icosidodecahedron? Describe the polygons that meet at each vertex.

Some of the Archimedean solids can be made by cutting off the corners of a regular polyhedron, or Platonic solid. Follow the steps below to make a truncated cube.

a. Draw a cube (or make a cube out of modeling clay).

b. Visualize a tilted plane that intersects the top, front, and right faces of the cube such that the cross section is an equilateral triangle. Th is plane should be closer to the corner of the cube than the midpoint of the edges. Draw this cross section (or cut off the corner of the cube of modeling clay).

c. Repeat step (b) with the other corners of the cube (or the clay). All cross sections should be congruent.

2. What are the regular polygons that make up the faces of a truncated cube? Describe the polygons that meet at each vertex.

3. Suppose that the cross sections have been formed by planes that cut off the corners of the cube and intersect the midpoints of the edges. Draw or model the solid that would be formed. What are the regular polygons that make up the faces of this solid?

4. Other Archimedean solids that can be made by cutting off the corners of a Platonic solid include the truncated tetrahedron and the truncated octahedron. Follow a set of steps similar to those above to create these solids. Th en describe the faces of each.

Pentagons and triangles; at each vertex two regular pentagons and two equilateral triangles meet.

Octagons and triangles; at each vertex two octagons and a triangle meet.

squares and triangles

truncated tetrahedron: hexagonal and triangular faces; truncated octahedron: hexagonal and square faces


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11-1 Reteaching


A polyhedron is a three-dimensional fi gure with faces that are polygons. Faces intersect at edges, and edges meet at vertices.

Faces, vertices, and edges are related by Euler’s Formula: F 1 V 5 E 1 2.

For two dimensions, such as a representation of a polyhedron by a net, Euler’s Formula is F 1 V 5 E 1 1. (F is the number of regions formed by V vertices linked by E segments.)

Problem

What does a net for the doorstop at the right look like? Label the net with its appropriate dimensions.

Exercises

Complete the following to verify Euler’s Formula.

1. On graph paper, draw three other nets for the polyhedron shown above. Let each unit of length represent 14 in.

2. Cut out each net, and use tape to form the solid fi gure.

3. Count the number of vertices, faces, and edges of one of the fi gures.

4. Verify that Euler’s Formula, F 1 V 5 E 1 2, is true for this polyhedron.

Draw a net for each three-dimensional fi gure.

5. 6.

3 in.

4 in.

8 in.

5 in.

Doorstop

3 in.

4 in.4 in.

8 in.5 in.5 in.

5 in.

5 in.8 in.

3 in.

5 in.4 in.

3 in.

4 in.

Sample:

Samples:

Check students’ work.

6 vertices, 5 faces, 9 edges

F 1 V 5 E 1 2, 5 1 6 5 9 1 2, 11 5 11

3 in.

8 in.

5 in.4 in.

3 in.

4 in.

3 in.

4 in.

3 in. 3 in.

8 in.

5 in.

4 in.

3 in.

5 in. 5 in.


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A cross section is the intersection of a solid and a plane. Cross sections can be many diff erent shapes, including polygons and circles.

Th e cross section of this solid and this plane is a rectangle. Th is cross section is a horizontal plane.

To draw a cross section, visualize a plane intersecting one face at a time in parallel segments. Draw the parallel segments, then join their endpoints and shade the cross section.

Exercises

Draw and describe the cross section formed by intersecting the rectangular prism with the plane described.

7. a plane that contains the vertical line of symmetry

8. a plane that contains the horizontal line of symmetry

9. a plane that passes through the midpoint of the top left edge, the midpoint of the top front edge, and the midpoint of the left front edge

10. What is the cross section formed by a plane that contains a vertical line of symmetry for the fi gure at the right?

11. Visualization What is the cross section formed by a plane that is tilted and intersects the front, bottom, and right faces of a cube?

11-1 Reteaching (continued)


Answers may vary. a rectangle or a square

a rectangle

an isosceles triangle

Answers may vary. a hexagon or a rectangle

a triangle


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Name Class Date

11-2 Additional Vocabulary SupportSurface Areas of Prisms and Cylinders

Complete the vocabulary chart by fi lling in the missing information.

Word or Word Phrase

Defi nition Picture or Example

altitude of a prism or cylinder

A perpendicular segment that joins the planes of the bases of a prism or cylinder is the altitude of a prism or cylinder.

AF , for example

bases of a prism or cylinder

1.

height of a prism or cylinder

Th e height of a prism or cylinder is the length of an altitude of the solid. It is the distance between the solid’s bases.

2.

lateral area of a prism or cylinder

3. L.A. 5 ph (prism)

L.A. 5 2prh or pdh (cylinder)

surface area of a prism or cylinder

Th e surface area of a prism or cylinder is the sum of the lateral area and the areas of the two bases.

4.

oblique prism or oblique cylinder

An oblique prism or oblique cylinder is a prism or cylinder with lateral faces and bases that are not perpendicular.

5.

right prism or right cylinder

6.

AB

C

DF

E

bases

The congruent, parallel faces on a prism or a cylinder are the bases of a prism or cylinder.

The lateral area of a prism or cylinder is the sum of the areas of the lateral (side) faces of a prism or cylinder.

A right prism or right cylinder is a prism or cylinder with lateral faces and bases that are perpendicular.

S.A. 5 L.A. 1 2B (prism)S.A. 5 2πrh 1 2πr2 (cylinder)

h


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11-2 Think About a Plan Surface Areas of Prisms and Cylinders

Reasoning Suppose you double the radius of a right cylinder. a. How does that aff ect the lateral area? b. How does that aff ect the surface area? c. Use the formula for surface area of a right cylinder to explain why the

surface area in part (b) was not doubled.


1. What is the formula for the lateral area of a right cylinder?

2. What is the formula for the surface area of a right cylinder?


3. How does doubling the radius aff ect the formulas for the lateral and surface areas? In the formula for the surface area, where do you need to be most careful?

4. How do you compare the new formulas you get after doubling the radius in the original formulas?

Getting an Answer

5. Write the formula for the new lateral area after the radius has been doubled. Compare this to the original formula for the lateral area. What eff ect does doubling the radius have?

6. Write the formula for the new surface area after the radius has been doubled. Compare this to the original formula for the surface area. What eff ect does doubling the radius have?

L.A. 5 2πrh

S.A. 5 2πrh 1 2πr2

Replace r with 2r everywhere it appears in each formula; in the formula for

surface area, be careful to apply the exponent of 2 to 2r, not just r.

Factor each formula or divide the new formula by the old formula.

Original formula: S.A. 5 2πrh 1 2πr2; the formula after doubling the radius is

New S.A. 5 2π(2r)h 1 2π(2r)2 5 4πrh 1 8πr2; the area of the base has been

quadrupled. So, the surface area has more than doubled.

Original formula: L.A. 5 2πrh; the formula after doubling the radius is

New L.A. 5 2π(2r)h 5 4πrh; because New L.A.L.A. 5 4πrh

2πrh 5 2, doubling the radius

doubles the lateral area.


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Surface Areas of Prisms and Cylinders

Use a net to fi nd the surface area of each prism. Round your answer to the nearest whole number.

1. 2.

3. a. Classify the prism at the right.

b. Th e bases are regular pentagons. Find the lateral area of the prism.

c. Th e area of each is 43 cm2. Find the sum of their areas.

d. Find the surface area of the prism.

Use formulas to fi nd the lateral area and surface area of each prism. Round your answer to the nearest whole number.

4. 5. 6.

Find the lateral area of each cylinder to the nearest whole number.

7. 8. 9.

10. A box of cereal measures 8 in. wide, 11 in. high, and 2 in. deep. If all surfaces are made of cardboard and the total amount of overlapping cardboard in the

box is 7 in.2, how much cardboard is used to make the cereal box?

11. Judging by appearances, what is the surface area of the solid shown at the right? Show your answer to the nearest whole number.

10 ft

4 ft4 ft

3 in.3 in.

10 in.

5 cm

11 cm

7 m

7 m

7 m

8.1

12

2.8

4 4

3

6 in.

10 in.

10 in.

24 in.

r 2 m

10 m 11 m

r 5m r 7 m

22 m

10 cm

6 cm

15 cm

192 ft2 111 in.2

pentagonal prism

275 cm2

86 cm2

361 cm2

196 m2; 294 m2 227; 260 624 in.2; 681 in.2

126 m2

346 m2 968 m2

259 in.2

1082 cm2


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Find the surface area of each cylinder in terms of π.

12. 13.

14. a. A cylindrical container of paint with radius 6 in. is 15 in. tall. If all of the surfaces except the top are made of metal, how much metal is used to make the container? Assume the thickness of the metal is negligible. Show your answer to the nearest square inch.

b. If the top of the paint container is made of plastic, how much plastic is used to make the top? Assume the thickness of the plastic is negligible. Show your answer to the nearest square inch.

15. a. Reasoning Suppose that a cylinder has a radius of r units and a height of 2r units. Th e lateral area of the cylinder is 64p square units. What is the value of r?

b. What is the surface area of the cylinder? Round your answer to the nearest square unit.

Visualization Suppose you revolve the plane region completelyabout the given line to sweep out a solid of revolution. Describethe solid and fi nd its surface area in terms of π.

16. the x-axis

17. the y-axis

18. the line x 5 3

19. the line y 5 2

20. An artist creates a right prism whose bases are regular decagons. He wants to paint the surface of the prism. One can of paint can cover 32 square feet. How many cans of paint must he buy if the height of the prism is 11 ft and the length of each side of the decagon is 2.4 ft? Th e area of a base is approximately 89 ft2.

21. Open-Ended Draw a cylinder with a surface area of 136p cm2.

32 cm

d 15 cmr = 10 in.

21 in.

y

x12345

1 2 3 4 5o



592.5π cm2620π in.2

679 in.2

113 in.2

4 units

302 square units

cylinder; 56π square units




Check students’ drawings. Sample: radius of bases 5 2 cm and height 5 32 cm, or radius of bases 5 4 cm and height 5 13 cm

14 cans


15

Name Class Date



Use a net to fi nd the surface area of each prism.

1. 2.

3. a. Classify the prism at the right.

b. Find the lateral area of the prism.

c. Th e bases are regular hexagons. Th e area of each is about 41.6 cm2. Find the sum of their areas.

d. Find the surface area of the prism.

Use formulas to fi nd the surface area of each prism. Round your answer to the nearest whole number.

4. To start, use the formula for the lateral area of a prism, then fi nd the perimeter of the base trapezoid.

L.A. 5 ph

p 5u 1u 1u 1u5u m

5. 6.

7. A box measures 10 in. wide, 12 in. high, and 14 in. deep. If all surfaces are made of cardboard, how much cardboard is used to make the box?

8. An artist creates a right prism whose bases are regular pentagons. He wants to paint the lateral surfaces of the prism. One can of paint can cover 30 ft2. How many cans of paint must he buy if the height of the prism is 15 ft and the length of each side of the pentagon is 5 ft?

5 m2 m

10 m6 ft

4 ft5 ft

4 cm4 cm

9 cm

4 cm4 cm 4 c

m

4 cm

30 m6 m5 m

3 m

7 m

4 m

5 m

5 m5 m

4.8 ft

2.2 ft

1.9 ft

160 m2

150 m2

666 m2

84 ft2

48 ft2

right hexagonal prism

856 in.2

216 cm2

13 cans

83.2 cm2

3 5 6 7

21

299.2 cm2


16

Name Class Date

Find the surface area of each cylinder in terms of π.

9. To start, use the formula for the surface area of the cylinder, then identify the variables and any given values.

S.A. 5 2prh 1 2pr2

r 5ucm, h 5ucm

10. 11.

Find the lateral area of each cylinder to the nearest whole number.

12. To start, use the formula for the lateral area of the cylinder, then identify the variables and any given values.

L.A. 5 pdh

h 5 6 in., d 5 2 ?u in. 5uin.

13. 14.

15. Reasoning A cylinder has a height that is 2 times as large as its radius. Th e lateral area of the cylinder is 16p square units.

a. What is the length of the radius of the cylinder?

b. What is the height of the cylinder?

c. What is the surface area of the cylinder? Round your answer to the nearest square unit.

16. Reasoning A triangular prism and a rectangular prism both have bases that are regular polygons with sides 2 units long. Which has a greater surface area? Explain.

12 cm

4 cm

20 mmd 18 mm

11 ft

4 ft

6 in.3 in.

d 12 m

20 m

12 cm

3.5 cm



128π cm2

754 m2

522π mm2

132 cm2

330π ft2

The surface area of a rectangular prism is greater. It has a greater perimeter, and its bases have greater areas.

113 in.2

2 units

3 6

4 12

4 units

75 square units


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11-2 Standardized Test Prep Surface Areas of Prisms and Cylinders

Multiple Choice


1. What is the lateral surface area of a cube with side length 9 cm?

72 cm2 324 cm2 405 cm2 486 cm2

2. What is the surface area of a prism whose bases each have area 16 m2 and whose lateral surface area is 64 m2?

80 m2 96 m2 144 m2 160 m2

3. A cylindrical container with radius 12 cm and height 7 cm is covered in paper. What is the area of the paper? Round to the nearest whole number.

528 cm2 835 cm2 1055 cm2 1432 cm2

For Exercises 4 and 5, use the prism at the right.

4. What is the surface area of the prism?

283.8 m2 325.4 m2

292.4 m2 407 m2

5. What is the lateral surface area of the prism?

283.8 m2 292.4 m2 325.4 m2 407 m2

For Exercises 6 and 7, use the cylinder at the right.

6. What is the lateral surface area of the cylinder?

12p cm2 216p cm2

18p cm2 288p cm2

7. What is the surface area of the cylinder?

12p cm2 18p cm2 216p cm2 288p cm2

8. Th e height of a cylinder is three times the diameter of the base. Th e surface area of the cylinder is 126p ft2. What is the radius of the base?

3 ft 6 ft 9 ft 18 ft

Short Response

9. What are the lateral area and the surface area of the prism?

5.5 m

3 m17.2 m

18 cm

d 12 cm

25 in.16 in.

12 in.

B

G

D

H

B

H

D

F

[2] L.A. 5 1200 in.2; S.A. 5 1392 in.2 [1] one of two answers correct [0] no correct response given


18

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11-2 Enrichment Surface Areas of Prisms and Cylinders

Constructing Rectangular BoxesGiven a rectangular sheet of material, such as paper or metal, it is possible to cut out squares and rectangles and reassemble the result into a right rectangular prism. For example, the sheet of paper pictured in the diagram is 8 in. by 12 in.

Use the fi gure at the right for Exercises 1–15.

1. What is the area of the rectangle?

2. Suppose that two 2-in. squares are cut out as indicated along the dotted lines. What is the total area of the two squares?

3. Th ese squares are to be used as bases of a right rectangular prism, and the remaining material is to be used to construct the lateral faces. After the bases have been cut out from the sheet, how much area remains?

4. How many lateral faces will the rectangular prism have?

5. What must be the area of each face?

6. What must be the height of the rectangular prism?

7. What is the surface area of the rectangular prism?

Now, suppose that each side of the square base is s in.

8. What is the total area of the bases of the rectangular prism that will be constructed?

9. How much area remains?

10. How many lateral faces will the rectangular prism have?

11. What must be the area of each face?

12. One length of each face is known because it is also the side of either the top or bottom. What is its length?

13. What must be the height of the rectangular prism?

14. What is the surface area of the rectangular prism?

15. Predict the surface area of the rectangular prism created if two 4-in. squares are cut out of the rectangle above and used as the bases. Explain.

96 in.2

8 in.2

88 in.2

22 in.2

11 in.

96 in.2

96 in.2

96 in.2; no matter what the size of the base is, the surface area of the rectangular prism will always be 96 in.2 as long as the entire piece of paper is used.

(2s2) in.2

(96 2 2s2) in.2

(24 2 0.5s2) in.2

4

4

s in.

(24s 2 0.5s) in.


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11-2 Reteaching Surface Areas of Cylinders and Prisms

A prism is a polyhedron with two congruent parallel faces called bases. Th e non-base faces of a prism are lateral faces. Th e dimensions of a right prism can be used to calculate its lateral area and surface area.

Th e lateral area of a right prism is the product of the perimeter of the base and the height of the prism.

L.A. 5 ph

Th e surface area of a prism is the sum of the lateral area and the areas of the bases of the prism.

S.A. 5 L.A. 1 2B

Problem

What is the lateral area of the regular hexagonal prism?

L.A. 5 ph

p 5 6(4 in.) 5 24 in. Calculate the perimeter.

L.A. 5 24 in. 3 13 in. Substitute.

L.A. 5 312 in.2 Multiply.

Th e lateral area is 312 in.2.

Problem

What is the surface area of the prism?

S.A. 5 L.A. 1 2B

p 5 2(7 m 1 8 m) Calculate the perimeter.

p 5 30 m Simplify.

L.A. 5 ph

L.A. 5 30 m 3 30 m Substitute.

L.A. 5 900 m2 Multiply.

B 5 8 m 3 7 m Find base area.

B 5 56 m2 Multiply.

S.A. 5 L.A. 1 2B

S.A. 5 900 m2 1 2 3 56 m2 Substitute.

S.A. 5 1012 m2 Simplify.

Th e surface area of the prism is 1012 m2.

4 in.

13 in.

8 m7 m

30 m


20

Name Class Date

A cylinder is like a prism, but with circular bases. For a right cylinder, the radius of the base and the height of the cylinder can be used to calculate its lateral area and surface area.

Lateral area is the product of the circumference of the base (2pr) and the height of the cylinder. Surface area is the sum of the lateral area and the areas of the bases (2pr2).

L.A. 5 2prh or pdh S.A. 5 2prh 1 2pr2

Problem

Th e diagram at the right shows a right cylinder. What are the lateral area and surface area of the cylinder?

L.A. 5 2prh or pdh

L.A. 5 2p 3 4 in. 3 9 in. Substitute for r and h.

L.A. 5 72p in.2 Multiply.

Th e lateral area is 72p in.2.

S.A. 5 2prh 1 2pr2

S.A. 5 2p 3 4 in. 3 9 in. 1 2p 3 (4 in.)2 Substitute for r and h.

S.A. 5 72p in.2 1 32p in.2 Multiply.

S.A. 5 104p in.2 Add.

Th e surface area is 104p in.2.

Exercises

Find the lateral area and surface area of each fi gure. Round your answers to the nearest tenth, if necessary.

1. 2. 3.

4. A cylindrical carton of raisins with radius 4 cm is 25 cm tall. If all surfaces except the top are made of cardboard, how much cardboard is used to make the raisin carton? Round your answer to the nearest square centimeter.

r 4 in.

9 in.

18 cm

5 cm

12 in.

12 in.

12 in.

3 m

5 m10 m


Surface Areas of Cylinders and Prisms

565.5 cm2; 722.6 cm2

679 cm2

576 in.2; 864 in.2

138.3 m2; 153.3 m2


21

Name Class Date

11-3 Additional Vocabulary SupportSurface Areas of Pyramids and Cones

Use the list below to complete the web.

Base is a circle. Base is a polygon. L.A. 5 12 p<

L.A. 5 12 ? 2πr< pyramid S.A 5 L.A 1 B

cone

Lateral area and base area are used to fi nd surface area.

half the product of the slant height and perimeter/circumference of the base

pyramid

Base is a polygon.

L.A. 5 12 p< L.A. 5 1

2 ? 2πr<S.A. 5 L.A. 1 B

Base is a circle.


22

Name Class Date

11-3 Think About a PlanSurface Areas of Pyramids and Cones

Find the lateral area of the cone to the nearest whole number.


1. What is the formula for the lateral area of a cone?

2. How are the two variables in this formula defi ned?

3. What two pieces of information are given in the fi gure of the cone?


4. How can you use the given information to fi nd the radius?

5. How can you use the given information and the radius to fi nd the slant height?

Getting an Answer

6. What is the radius?

7. What is the slant height of the cone?

8. What is the lateral area of the cone?

4 m

4.5 m

The radius, r, is the distance from the center of the circle to a point on the

circle. The slant height, <, is the distance from the vertex of the cone to a

point on the circle.

the diameter and the height

Divide the length of the diameter by 2.

Substitute the height and the radius into the Pythagorean Theorem, < 5"r2 1 h2.

L.A. 5 πr<

r 5 12 ? 4 m 5 2 m

< 5"22 1 (4.5)2 5"24.25 m

L.A. 5 π ? 2 ? "24.25 5 2"24.25π N 31 m2


23

Name Class Date


Surface Areas of Pyramids and Cones

Find the lateral area of each pyramid to the nearest whole number.

1. 2.

Find the surface area of each pyramid to the nearest whole number.

3. 4. 5.

Find the lateral area of each cone to the nearest whole number.

6. 7.

Find the surface area of each cone in terms of π.

8. 9. 10.

11. Th e surface area of a cone is 16.8p in.2. Th e radius is 3 in. What is the slant height?

12. Th e lateral area of a cone is 155.25p m2. Th e slant height is 13.5 m. What is the radius?

13. Th e roof of a clock tower is a pentagonal pyramid. Each side of the base is 7 ft long. Th e slant height is 9 ft. What is the area of the roof?

14. Write a formula to show the relationship between lateral area and the length of a side of the base (s) and slant height in a square pyramid.

6 m

10 m

5 m

6.9 m

9 m

12 m2 cm

30 cm

2 cm

5 m

10 m

4 cm

6 cm

13 cm

10 cm

12 cm

8 cm

12 cm

16 cm

10 cm

12 cm

120 m2

297 m2 128 m2

124 cm2

38 cm2

96π cm2

2.6 in.

11.5 m

90π cm2384π cm2

244 cm2

104 m2

157.5 ft2

L.A. 5 12 p< 5 1

2(4s)< 5 2s<


24

Name Class Date

Find the surface area to the nearest whole number.

15. 16.

17. Write a formula to show the relationship between surface area and the length of a side of the base (s) and slant height in a square pyramid.

Th e length of a side (s) of the base, slant height (<), height (h), lateral area (L.A.), and surface area (S.A.) are measurements of a square pyramid. Given two of the measurements, fi nd the other three to the nearest tenth.

18. s 5 9 cm, / 5 14.5 cm

19. L.A. 5 1542.4 m2, S.A. 5 2566.4 m2

20. h 5 9 cm, / 5 11.4 cm

Visualization Suppose you revolve the plane region completely about the given line to sweep out a solid of revolution. Describe the solid. Th en fi nd its surface area in terms of π. Round to the nearest tenth.

21. about the y-axis

22. about the x-axis

23. about the line x 5 2

24. about the line y 5 5

25. Open-Ended Draw a cone with a lateral area of 28p cm2. Label its dimensions. Th en fi nd its surface area.

6 cm

10 cm10 cm14 cm

10 cm

26 in.10 in.

8

40 in.

424 O

2

4

2

4

x

y

2



553 cm2

S.A. 5 2s< 1 s2

h 5 13.8 cm; L.A. 5 261 cm2; S.A. 5 342 cm2

s 5 14.0 cm; L.A. 5 319.2 cm2; S.A. 5 515.2 cm2

cone; S.A. 5 14.8π units2; 46.5 units2

cone; S.A. 5 51.9π units2; 163.0 units2

a cylinder with a cone cut out; S.A. 5 34.8π units2; 109.3 units2

a cylinder with a cone cut out; S.A. 5 71.9π units2; 225.9 units2

Check students’ drawings. Sample: radius of base = 4 cm and slant height = 7 cm; 44π cm2

h 5 18 m; s 5 32 m; < 5 24.1 m

3644 in.2


25

Name Class Date



Find the surface area of each pyramid to the nearest whole number.

1. To start, use the formula for surface area of the pyramid, then identify the variables and any given values.

S.A. 5 12 p/ 1 B

p 5 4 ?u 5u in.

/ 5u in.

B 5u ?u 5u in.2

2. 3.

Find the lateral area of each pyramid to the nearest whole number.

4. 5.

6. Th e fi gure at the right has one base and eight lateral faces.Find its surface area to the nearest whole number.

7. Th e roof of a clock tower is a square pyramid. Each side of the base is 16 ft long. Th e slant height is 22 ft. What is the lateral area of the roof?

8. Reasoning Write a formula to show the relationship between surface area and the length of a side of the base (s) and slant height in a square pyramid.

Th e length of a side (s) of the base, slant height (<), height (h), lateral area (L.A.), and surface area (S.A.) are measurements of a square pyramid. Given two of the measurements, fi nd the other three to the nearest tenth.

9. s 5 16 cm, / 5 10 cm

10. L.A. 5 624 m2, S.A. 5 1200 m2

11. h 5 7 cm, / 5 25 cm

7 in.

11 in.

6 m

6 m

8 m

5 m

12 m

4 m

6 m

10 in.

15 in.

8 m

8 m

5 m

6 m

203 in.2

132 m2 145 m2

60 m2 316 in.2

320 m2

704 ft2

S.A. 5 2s< 1 s2

h 5 6 cm; L.A. 5 320 cm2; S.A. 5 576 cm2

s 5 48 cm; L.A. 5 2400 cm2; S.A. 5 4704 cm2

s 5 24 m; < 5 13 m; h 5 5 m

7

7 7 49

28

11


26

Name Class Date

Find the surface area of each cone in terms of π.

12. To start, use the formula for surface area of the pyramid, then identify the variables and any given values.

S.A. 5 pr/ 1 B

r 5u mm

/ 5u mm

B 5 p ?u25u mm2

13. 14.

Find the lateral area of each cone to the nearest whole number.

15. 16.

17. Find the surface area of the fi gure at the right to the nearest whole number. (Hint: Add the base, the lateral area of the cylinder, and the lateral area of the cone.)

18. Th e lateral area of a cone is 60p m2. Th e slant height is 15 m. What is the radius?

19. Th e surface area of a cone is 55p cm2. Th e radius is 5 cm. What is the slant height?

4 mm

8 mm

10 cm

14 cm10 ft

6 ft

35 in.

30 in.12 m

5 m

7 mm

8 mm

6 mm



4

4

8

16π48π mm2

95π cm2

3299 in.2

547 mm2

4 m

6 cm

204 m2

39π ft2


27

Name Class Date

11-3 Standardized Test Prep Surface Areas of Pyramids and Cones

Multiple Choice


1. What is the lateral surface area of a square pyramid with side length 11.2 cm and slant height 20 cm?

224 cm2 448 cm2 896 cm2 2508.8 cm2

2. What is the lateral surface area of a cone with radius 19 cm and slant height 11 cm?

19p cm2 30p cm2 200p cm2 209p cm2

3. What is the lateral area of the square pyramid, to the nearest whole number?

165 m2 330 m2

176 m2 351 m2

4. What is the surface area of the cone, to the nearest whole number?

221 cm2 304 cm2

240 cm2 620 cm2

5. What is the surface area of a cone with diameter 28 cm and height 22 cm in terms of p?

196p cm2 365p cm2 561.1p cm2 2202.8p cm2

Extended Response

6. What are the perimeter of the base, slant height, lateral area, and surface area for the square pyramid, to the nearest tenth of a meter or square meter?

11 m

15 m

9 cm

12.5 cm

6 m

12 m

B

I

D

G

C

[4] p 5 24 m; < 5 12.4 m; L.A. 5 148.4 m2; S.A. 5 184.4 m2

[3] any three of the four values correctly given[2] any two of the four values correctly given [1] any one correct value given[0] no correct responses given


28

Name Class Date

11-3 EnrichmentSurface Areas of Pyramids and Cones

Frustum of a SolidA frustum of a pyramid or cone is the fi gure made when the tip of the pyramid or cone is cut off by a cross section that is perpendicular to the height. In a frustum of a pyramid or cone, there are two bases, an upper base and a lower base.

Th e fi gure at the upper right is a frustum of a square pyramid for which h is the height, a is the length of a side of the lower base, b is the length of a side of the upper base, and / is the slant height.

Th e fi gure at the lower right is a frustum of a cone for which h is the height, R is the radius of the lower base, r is the radius of the upper base, and / is the slant height.

Use the frustum of the regular square pyramid above to derive a surface area formula.

1. What is the area of its lower base?

2. What is the area of its upper base?

3. What is the shape of each lateral face?

4. What is the area of each lateral face?

5. What is the lateral area of the fi gure?

6. What is the formula for the surface area of a frustum of a square pyramid?

Use the frustum of the cone above to derive a surface area formula.

7. What is the area of its lower base?

8. What is the area of its upper base?

9. What is the lateral area of the fi gure? (Hint: Consider the formula for the lateral area of a cone.)

10. What is the formula for the surface area of a frustum of a cone?

Find the surface area of each fi gure below. Round your answers to the nearest tenth.

11. 12. 13. 14.

hb

a

,

h ,

r

R

5 cm

6 cm

6 cm

4 cm

11 in

.

15 in.

4 in.

7 in.

8 ft 10 ft

12 ft

9 ft

22 m

20 m

18 m

12 m

a2

152 cm2

722.6 in.2 506.6 ft21952 m2

b2

πR2

πr2

πR< 1 πr<

π(R2 1 r2 1 R< 1 r<)

a2 1 b2 1 2< (a 1 b)

trapezoid

2<(a 1 b)

12<(a 1 b)


29

Name Class Date

11-3 ReteachingSurface Areas of Pyramids and Cones

A pyramid is a polyhedron in which the base is any polygon and the lateral faces are triangles that meet at the vertex. In a regular pyramid, the base is a regular polygon. Th e height is the measure of the altitude of a pyramid, and the slant height is the measure of the altitude of a lateral face. Th e dimensions of a regular pyramid can be used to calculate its lateral area (L.A.) and surface area (S.A.).

L.A. 5 12p/, where p is the perimeter of the base and / is slant height of

the pyramid.

S.A. 5 L.A. 1 B, where B is the area of the base.

Problem

What is the surface area of the square pyramid to the nearest tenth?

S.A. 5 L.A. 1 B

L.A. 5 12 p/ Find lateral area fi rst.

p 5 4(4 m) 5 16 m Find the perimeter.

/2 5 "22 1 102 5 "104 Use the Pythagorean Theorem.

/ < 10.2

L.A. 5 12 (16m)(10.2) 5 81.6 m2 Substitute to fi nd L.A.

B 5 (4 m)(4 m) 5 16 m2 Find area of the base.

S.A. 5 81.6 m2 1 16 m2 5 97.6 m2 Substitute to fi nd S.A.

Th e surface area of the square pyramid is about 97.6 m2.

ExercisesUse graph paper, scissors, and tape to complete the following.

1. Draw a net of a square pyramid on graph paper.

2. Cut it out, and tape it together.

3. Measure its base length and slant height.

4. Find the surface area of the pyramid.

In Exercises 5 and 6, round your answers to the nearest tenth, if necessary.

5. Find the surface area of a square pyramid with base length 16 cm and slant height 20 cm.

6. Find the surface area of a square pyramid with base length 10 in. and height 15 in.

s

,

4 m

10 m

Sample:

Sample: base 5 3 cm, slant height 5 4 cm


Sample: 33 cm2

896 cm2

416.2 in.2


30

Name Class Date

A cone is like a pyramid, except that the base of a cone is a circle. Th e radius of the base and cylinder height can be used to calculate the lateral area and surface area of a right cone.

L.A. 5 pr /, where r is the radius of the base and / is slant height of the cone.

S.A. 5 L.A. 1 B, where B is the area of the base (B 5 pr2).

Problem

What is the surface area of a cone with slant height 18 cm and height 12 cm?

Begin by drawing a sketch.

Use the Pythagorean Th eorem to fi nd r, the radius of the base of the cone.

r2 1 122 5 182

r2 1 144 5 324

r2 5 180

r < 13.4

Now substitute into the formula for the surface area of a cone.

S.A. 5 L.A. 1 B

5 pr/ 1 pr2

5 p(13.4)(18) 1 180p

< 1323.2

Th e surface area of the cone is about 1323.2 cm2.

In Exercises 7–10, round your answers to the nearest tenth, if necessary.

7. Find the surface area of a cone with radius 5 m and slant height 15 m.

8. Find the surface area of a cone with radius 6 ft and height 11 ft.

9. Find the surface area of a cone with radius 16 cm and slant height 20 cm.

10. Find the surface area of a cone with radius 10 in. and height 15 in.

,

r

18 cm12 cm

r



314.2 m2

349.3 ft2

1809.6 cm2

880.5 in.2


31

Name Class Date

11-4 Additional Vocabulary SupportVolumes of Prisms and Cylinders

Problem

What is the volume of the prism at the right? Justify and explain your work.

Explain Work Justify

First, write the formula for volume of a prism.

V 5 Bh formula for volume of a prism

Second, substitute the value of the height.

V 5 B(2) substitution, h 5 2 cm

Next, substitute an expression for the area of the base.

V 5 (6.2 ? 6)(2) The base is a rectangle with side lengths 6 cm and 6.2 cm.

Finally, find the product. The volume of the rectangular prism is 74.4 cm3.

V 5 74.4 Multiply.

ExerciseWhat is the volume of the prism at the right? Justify and explain your work.

Explain Work Justify

First,V 5 Bh

________________________

________________________

Second,V 5 B(9)

Next, V 5 1

2 (3 ? 4.2)(9)________________________

________________________

Finally, V 5 56.7

6 cm2 cm 6.2 cm

4.2 cm

3 cm9 cm

Solution74.4 cm3

Solution56.7 cm3

write the formula for volume of a prism.

substitute the value of the height.

find the product. The volume of the rectangular prism is 56.7 cm3.

substitute an expression for the area of the base.

formula for volume of

a prism

substitution, h 5 9 cm

Multiply.

The base is a right triangle

with legs 3 cm and 4.2 cm.


32

Name Class Date

11-4 Think About a Plan Volumes of Prisms and Cylinders

Swimming Pool Th e approximate dimensions of an Olympic-size swimming pool are 164 ft by 82 ft by 6.6 ft.

a. Find the volume of the pool to the nearest cubic foot.

b. If 1 ft3 < 7.48 gallons, about how many gallons does the pool hold?


1. What are the dimensions of the pool?

2. How long is the pool? How wide is the pool? How deep is the pool?

3. What are the dimensions of the base of the pool?

4. How do you convert cubic feet to gallons?


5. How would you use the dimensions of the pool to fi nd its volume?

6. How would you use the volume of the pool to fi nd the number of gallons the pool holds?

Getting an Answer

7. What is the area of the base of the pool?

8. What is the volume of the pool?

9. About how many gallons of water does the pool hold?

164 ft by 82 ft by 6.6 ft

164 ft long; 82 feet wide; 6.6 ft deep

164 ft-by-82 ft

Multiply the number of cubic feet by 7.48.

Multiply the volume of the pool by 7.48.

B 5 < ?w 5 164 ?82 5 13,448 ft2

V 5 B ?h 5 13,448 ?6.6 5 88,756.8 ft3

The pool holds about 88,756.8 3 7.48 N 663,901 gallons.

First fi nd the area of the base of the pool by substituting the values

of the dimensions into the formula B 5 < ?w, then fi nd the volume by

substituting the values for B and h into the formula V 5 B ?h.


33

Name Class Date


Volumes of Prisms and Cylinders

Find the volume of each rectangular prism.

1. 2. 3.

4. 5. 6.

7. Th e base is a square, 4.5 cm on a side. Th e height is 5 cm.

8. Th e base is a rectangle with length 3.2 cm and width 4 cm. Th e height is 10 cm.

Find the volume of each triangular prism to the nearest tenth.

9. 10. 11.

12. Th e base is a right triangle with a leg of 12 in. and hypotenuse of 15 in. Th e height of the prism is 10 in.

13. Th e base is a 308-608-908 triangle with a hypotenuse of 10 m. Th e height of the prism is 15 m. Find the volume to the nearest tenth.

Find the volume of each cylinder in terms of π and to the nearest tenth.

14. 15. 16.

17. a right cylinder with a radius of 3.2 cm and a height of 10.5 cm

18. a right cylinder with a diameter of 8 ft and a height of 15 ft.

10.5 cm

12 cm

8 cm15 cm

3 cm

5 cm

6 cm

9.5 cm

2.5 cm

7.6 m

6 m

10 m

1.5 yd

3.5 yd

15.75 yd 9.8 in.

5.75 in.

1 in.

7 mm

18 mm

35 mm

19 cm

18 cm

18 cm

22 cm 25 m7 m

22 m

8.5 m

7 m8.5 cm

4.5 cm

2 mm

10 mm

1008 cm3

456 m3

2205 mm3

505.8π m3; 1588.9 m3

107.5π cm3; 337.8 cm3

240π ft; 754 ft3

172.1π cm3; 540.7 cm340π mm3; 125.7 mm3

3195.4 cm3 1925 m3

324.8 m3

540 in.3

56.35 in.3

128 cm3

82.6875 yd3

225 cm3

142.5 cm3

101.25 cm3


34

Name Class Date

Find the volume of each composite fi gure to the nearest whole number.

19. 20. 21.

Find the value of x to the nearest tenth.

22. Volume: 576 cm3 23. Volume: 980 mm3 24. Volume: 602.88 cm3

25. A cylindrical weather satellite has a diameter of 6 ft and a height of 10 ft. What is the volume available for carrying instruments and computer equipment, to the nearest tenth of a cubic foot?

26. A No. 10 can has a diameter of 15.5 cm and a height of 17.5 cm. A No. 2.5 can has a diameter of 9.8 cm and a height of 11 cm. What is the diff erence in volume of the two can types, to the nearest cubic centimeter?

27. Th e NCAA recommends that a competition diving pool intended for use with two 1-m springboards and two 3-m springboards, in addition to diving platforms set at 5 m, 7.5 m, and 10 m above the water, have a width of 75 ft 1 in., a length of 60 ft, and a minimum water depth of 14 ft 10 in. What is the minimum volume of water such a pool would hold in cubic yards, to the nearest whole number?

28. What is the volume of the solid fi gure formed by the net?

5 ft

10 ft6 cm

6 cm

3 cm

12 cm

8 cm

18 cm

4 in.

8 in.

10 in.18 in.

8 cm

8 cm

x

20 mm14 mm

x

12 cm

x

#10#2.5

4 m

8 m

10 m 4 m



1872 cm3

9.9 mm

2472 cm3

320 m3

2475 yd3

4.0 cm18 cm

1214 in.3

589 ft3

282.7 ft3


35

Name Class Date



Find the volume of each rectangular prism.

1. 2.

3. 4.

5. Th e base is a square, 9.6 cm on a side. Th e height is 6.2 cm.

6. Th e base is a rectangle with length 4.7 cm and width 7.5 cm. Th e height is 6.1 cm.

Find the volume of each triangular prism to the nearest tenth.

To start, use the formula 7. 8. for the volume of atriangular prism andthe formula for the base area of a triangle.

V 5 BH , B 5 12 bh

9. Th e base is a right triangle with a leg of 8 in. and hypotenuse of 10 in. Th e height of the prism is 15 in. (Hint: Use the Pythagorean Th eorem to fi nd the length of the other leg.)

10. Th e base is a 308-608-908 triangle with a hypotenuse of 14 m. Th e height of the prism is 11 m. Find the volume to the nearest tenth.

5 cm

4 cm

8 cm9 in.

5 in.

7 in.

6 m

15 m

7 m 6 yd 6 yd

6 yd

4 m

6 m

10 m

10 ft

16 ft

20 ft

160 cm3 315 in.3

630 m3 216 yd3

571.392 cm3

215.025 cm3

120 m3 1385.6 ft3

360 in.3

466.8 m3


36

Name Class Date

Find the volume of each cylinder in terms of π and to the nearest tenth.

To start, use the formula 11. 12. for the volume of acylinder, then identifythe variables and anygiven values.

V 5 pr2h

13. Th e radius of the right cylinder is 6.3 cm. Th e height is 14.5 cm.

14. Th e diameter of the right cylinder is 16 ft. Th e height is 7 ft.

Find the volume of each composite fi gure to the nearest whole number.

15. 16.

Find the volume of each fi gure to the nearest tenth.

17. 18.

19. A cylindrical weather satellite has a diameter of 10 ft and a height of 6 ft. What is the volume available for carrying instruments and computer equipment, to the nearest tenth of a cubic foot?

20. Can A has a diameter of 6 cm and a height of 6.5 cm. Can B has a diameter of 16 cm and a height of 11.5 cm. What is the diff erence in volume of the two can types, to the nearest cubic centimeter?

6 m

2 m

30 cm

14 cm

2 m

6 m

8 m6 in.

6 in.

14 in.

3 in.

8 mm

7 mm

10 mm

6 ft

3 ft

AB



24π m3; 75.4 m35880π cm3; 18,472.6 cm3

575.5π cm3; 1808.0 cm3

448π ft3; 1407.4 ft3

105 m3 630 in.3

560 mm3 169.6 ft3

2128 cm3

471.2 ft3


37

Name Class Date

11-4 Standardized Test Prep Volumes of Prisms and Cylinders

Gridded Response

Solve each exercise and enter your answer on the grid provided.

1. What is the volume in cubic inches of the prism?

2. What is the volume in cubic feet of the prism, rounded to the nearest cubic foot?

3. What is the volume in cubic inches of the cylinder, rounded to the nearest cubic inch?

4. What is x, if the volume of the cylinder is 768p cm3?

5. What is the volume in cubic inches of the solid fi gure, rounded to the nearest cubic inch?

Answers

1. 2. 3. 4. 5.

8 in.

5 in.

3 in.

3 ft

3 ft

3 ft 9 ft

9 in.

7 in.

8 cm

x cm

11 in.6 in.

12 in.17 in.

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38

Name Class Date

11-4 Enrichment Volumes of Prisms and Cylinders

To describe positions in space, you need a three-dimensional coordinate system. Th is system is set up using three perpendicular lines, called the x-axis, y-axis, and z-axis. Points in a three-dimensional coordinate system are called ordered triples, or (x, y, z). Point A is located at (3, 22, 5).

Find the volume of each prism graphed on a three-dimensional coordinate system.

1. 2.

3. 4.

Use the given ordered triples to graph each prism on a three-dimensional coordinate system. Th en fi nd the volume of each.

5. right rectangular prism: (3, 0, 0), (3, 5, 0), (0, 5, 0), (0, 0, 0), (3, 0, 6), (3, 5, 6), (0, 5, 6), (0, 0, 6)

6. right triangular prism: (5, 0, 0), (0, 25, 0), (0, 0, 0), (5, 0, 2), (0, 25, 2), (0, 0, 2)

6

6

2

4

6

66

4

4

4

2

44

6

2

z

x

y

(3, 2, 5) A

(0, 0, 2)(4, 0, 2)

(4, 0, 0)

(4, 6, 2)(4, 6, 0) (0, 6, 0)

(0, 6, 2)

z

x

y

(3, 0, 5)

(0, 0, 5)

(0, 4, 0)

(3, 0, 0) (0, 0, 0)

(0, 4, 5)

z

x

y

(4, 2, 0)

(4, 2, 4)

(0, 2, 4) (0, 2, 4)

(0, 2, 0)

(4, 2, 4)

(4, 2, 0)

z

x

y(0, 4, 0)

(2, 0, 3)

(0, 4, 3) (0, 1, 3)

(0, 1, 0)

(2, 0, 0)

z

x

y

48 units3

64 units3

90 units3

25 units3

15 units3

30 units3

5.

6.


39

Name Class Date

11-4 Reteaching Volumes of Prisms and Cylinders

Problem

Which is greater: the volume of the cylinder or the volume of the prism?

Volume of the cylinder: V 5 Bh

5 pr2 ? h

5 p(3)2 ? 12

< 339.3 in.3

Volume of the prism: V 5 Bh

5 s2 ? h

5 62 ? 12

5 432 in.3

Th e volume of the prism is greater.

Exercises

Find the volume of each object.

1. the rectangular prism part of the milk container

2. the cylindrical part of the measuring cup

Find the volume of each of the following. Round your answers to the nearest tenth, if necessary.

3. a square prism with base length 7 m and height 15 m

4. a cylinder with radius 9 in. and height 10 in.

5. a triangular prism with height 14 ft and a right triangle base with legs measuring 9 ft and 12 ft

6. a cylinder with diameter 24 cm and height 5 cm

12 in.

6 in.6 in.

6 in.

12 in.

7 cm7 cm

10 cm

6.3 cm7 cm

490 cm3

735 m3

2261.9 cm3

2544.7 in.3

756 ft3

77.175π or about 242.5 cm3


40

Name Class Date

Problem

What is the volume of the triangular prism?

Sometimes the height of a triangular base in a triangular prism is not given. Use what you know about right triangles to fi nd the missing value. Th en calculate the volume as usual.

hypotenuse 5 18 cm Given

short leg 5 9 cm 308-608-908 triangle theorem

long leg 5 9"3 cm 308-608-908 triangle theorem

Volume of prism: V 5 Bh

V 5 a12b (9)(9"3)(12)

V < 841.8 cm3

Th e volume of the triangular prism is about 841.8 cm3.

Exercises

Find the volume of each prism. Round to the nearest tenth.

7. 8. 9.

10. 11. 12.

Find the volume of each composite fi gure to the nearest tenth.

13. 14. 15.

12 cm

18 cm

30

60

45

455 cm

3 2 cm

25 in.

27 in.

9 in. 10 mm

16 mm

22 mm

3 in.

1 in.

3 m

2 m

7 m

6 in.

17 in.

6 in.6 in.

6 ft

3 ft2 ft

4 ft8 ft

10 in.

3 in.3 in.

2 in.

4 in.5 in.



22.5 cm3

7.1 in.3

76 ft3 96.8 in.3 111.4 in.3

265.0 in.342 m3

2833.8 in.3

1208.0 mm3


41

Name Class Date

11-5 Additional Vocabulary SupportVolumes of Pyramids and Cones

Th ere are two sets of note cards below that show how to fi nd the volume of a cone. Th e set on the left explains the thinking. Th e set on the right shows the work. Write the thinking and the steps in the correct order.

Think Cards Write Cards

Think Write

16

16

Step 1

Step 2

Step 3

First,

Second,

Next,

Then, Step 4

Step 5Finally,

Use a calculator.

Substitute the values in the formula.in the formula.

Simplify.

Round to the nearest cubic unit.cubic unit.

Use the formula for volume of a cone.

V N 1072

V 5 13pr

2h

V 5 13p(8)2(16)

V 5 34113p

V N 1072.330292

use the formula for volume of a cone.

V 5 13 πr2h

V 5 13π(8)2(16)

V 5 34113π

substitute the values in the formula.

simplify.

use a calculator. V N 1072.330292

V N 1072round to the nearest cubic unit.


42

Name Class Date

11-5 Think About a Plan Volumes of Pyramids and Cones

Writing Th e two cylinders pictured at the right are congruent. How does the volume of the larger cone compare to the total volume of the two smaller cones? Explain.


1. You are told that the two cylinders are congruent. What does that tell you about the cones?

2. You are asked to compare the volumes of the cones. Can you fi nd the exact volumes of the cones? In what other way can you compare their volumes?


3. What are the variables in the formula for the volume of a cone? What do they represent?

4. Let r represent the radius of the larger cone. What is the radius of the smaller cones in terms of r? Let h represent the height of the larger cone. What is the height of the smaller cones in terms of h? Explain.

Getting an Answer

5. What is the formula for the volume of the larger cone? What is the formula for the volume of the smaller cones?

6. How does the volume of the larger cone compare to the total volume of the smaller cones?

It tells you that the larger cone is twice as tall as the smaller cones.

No; you can compare their volumes using the formulas for the volumes of

the cones.

Because the radii are equal, the radius of the smaller cones is also r; because

the height of the smaller cones is one-half the height of the larger cone, the

height of the smaller cones is 12 h.

Because 13 πr 2

(12 h) 1 13 πr

2 (12 h) 5 1

3 πr 2h, the volume of the larger cone is equal

to the total volume of the two smaller cones.

The variables are the radius of the base, r, and the height of the cone, h.

larger cone: V 5 13 πr

2h; smaller cones: V 5 13 πr

2 (12 h)


43

Name Class Date


Volumes of Pyramids and Cones

Find the volume of each square pyramid. Round to the nearest tenth if necessary.

1. 2. 3.

Find the volume of each square pyramid, given its slant height. Round to the nearest tenth.

4. 5. 6.

7. Th e base of a pyramid is a square, 4.5 cm on a side. Th e height is 5 cm. Find the volume.

8. Th e base of a pyramid is a square, 3.2 cm on a side. Th e height is 10 cm. Find the volume to the nearest tenth.

Find the volume of each cone in terms of π and also rounded as indicated.

9. nearest cubic foot 10. nearest cubic meter 11. nearest cubic inch

12. Th e base has a radius of 16 cm and a height of 12 cm. Round to the nearest cubic centimeter.

13. Th e base has a diameter of 24 m and a height of 15.3 m. Round to the nearest cubic meter.

Find the volume to the nearest whole number.

14. 15. 16.

10 in.

8 in.8 in.

4 cm

6 cm

5 cm

3 cm3 cm

2.8 cm

1.2 cm

12.8 mm

13 mm

14.2 m

5 m

14 ft

24 ft12 m

8 m 15 in.

5 in.

12 cm

42 cm

36 cm

15 cm28 m

25 m

48 cm3

1.3 cm3 621.2 mm3

33.75 cm3

5542 cm3 5089 cm3 5131 m3

34.1 cm3

392π ft3; 1232 ft364π m3; 201 m3

31.25π in.3; 98 in.3

1024π cm3; 3217 cm3

734.4π m3; 2307 m3

116.5 m3

213.3 in. 3 15 cm3


44

Name Class Date

Find the volume of each fi gure to the nearest whole number.

17. 18. 19.

Algebra Find the value of x in each fi gure. Leave answers in simplest radical form. Th e diagrams are not to scale.

20. 21. 22.

23. One right circular cone is set inside a larger right circular cone. Th e cones share the same axis, the same vertex, and the same height. Find the volume of the space between the cones if the diameter of the inside cone is 6 in., the diameter of the outside cone is 9 in., and the height of both is 5 in. Round to the nearest tenth.

24. Some Native Americans still use tepees for special occasions and ceremonial purposes. Each group attending a family reunion, for example, might bring a small tepee, while using a larger tepee like the one pictured at the right for gathering together. Th e many poles form a rough cone with a circular base. What is the approximate volume of air in the tepee at the right, to the nearest cubic foot?

Visualization Suppose you revolve the plane region completely about the given line to sweep out a solid of revolution. Describe the solid. Th en fi nd its volume in terms of π.

25. the x-axis 26. the y-axis

27. the line x 5 3

28. the line y 5 22

9 cm

12 cm

15 cm

10 m

10 m

10 m

13 m

16 mm

8 mm

4 mm

16 mm

x

Volume 150015

15

x

Volume 8

6

x

14

9Volume 126

16.8 ft

19.5 ft

2 O

2

2

x

y



4835 cm3

20 2 6

1433 m3

58.9 in.3

1672 ft3

cone; 4π

cylinder with cone cut out; 12π

cone; 6π

cylinder with cone cut out; 8π

938 mm3


45

Name Class Date



Find the volume of each square pyramid. Round to the nearest tenth if necessary.

To start, use the formula for the volume of a pyramid. Th en fi nd the area of the base of the pyramid.

V 5 13 Bh

1. 2.

Find the volume of each square pyramid, given its slant height. Round to the nearest whole number.

To start, fi nd the height of the pyramid using the Pythagorean Th eorem. Th en use the formula for the volume of a pyramid.

3. 4.

5. Th e base of a pyramid is a square, 24 cm on a side. Th e height is 13 cm. Find the volume.

6. Th e base of a pyramid is a square, 14 cm on a side. Th e height of the pyramid is 25 cm. Find the volume to the nearest whole number.

Find the volume of each cone in terms of π and also rounded as indicated.

7. nearest cubic foot 8. nearest cubic inch

9. Th e base has a radius of 8 cm and a height of 5 cm. Round to the nearest cubic centimeter.

10. Th e base has a diameter of 20 m and a height of 12.6 m. Round to the nearest cubic meter.

15 cm

10 cm

7 cm

6 cm

32 cm

30 cm

5 m

8 m

7 ft

4.2 ft

42 in.

110 in.

750 cm398 cm3

8662 cm3 63 m3

2496 cm3

1633 cm3

41.16π ft3; 129 ft316,170π in.3; 50,800 in.3

106 23π cm3; 335 cm3

420π m3; 1319 m3


46

Name Class Date

Find the volume of each fi gure to the nearest whole number.

11. 12.

13. 14.

15. 16.

17. One right circular cone is set inside a larger right circular cone. Find the volume of the space between the cones if the diameter of the inside cone is 9 in., the diameter of the outside cone is 15 in., and the height of both is 8 in. Round to the nearest tenth.

18. Th e Pyramid of Khufu is a square pyramid which had a side length of about 230 m and a height of about 147 m when it was completed. Th e Pyramid of Khafre had a side length of about 215 m and a height of about 144 m when it was completed. What was the approximate diff erence in the volume of the two pyramids upon completion?

12 cm8 cm

12 m

10 m

8 cm

4 cm5 cm

6 cm

4 cm

8 cm

6 cm

7

7

710

6



302 cm3

224 cm3

114 unit3 302 unit3

301.6 in.3

373,300 m3

503 cm3

1257 m3


47

Name Class Date

11-5 Standardized Test Prep Volumes of Pyramids and Cones

Multiple Choice


1. What is the volume of the pyramid?

56 ft3 196 ft3

130 23 ft3 392 ft3

2. What is the volume of the cone, rounded to the nearestcubic inch?

72 in.3 905 in.3

226 in.3 2714 in.3

3. What is the volume of the fi gure?

15 cm3 45 cm3

33 cm3 54 cm3

4. What is the value of x, if the volume of the cone is 12p m3?

4 m 6 m

5 m 10 m

5. What is the diameter of a cone with height 8 m and volume 150p m3?

7.5 m 5"3 m 7.5"3 m 15 m

Short Response

6. Error Analysis A student calculates the volume of the given cone as approximately 2094 cm3. Explain the error in the student’s reasoning and fi nd the actual volume of the cone rounded to the nearest whole number.

7 ft8 ft

7 ft

12 in.

6 in.

3 cm

2 cm

3 cm3 cm

5 m

x

6 m

20 cm

20 cm

60

B

G

B

F

D

[2] The student uses slant height of the cone instead of the height to fi nd the volume. The actual volume is approximately 1814 cm3. [1] incorrect explanation or incorrect volume [0] no correct responses given


48

Name Class Date

11-5 Enrichment Volumes of Pyramids and Cones

It is possible to devise a formula for the volume of a regular pyramid that involves only the length of a side of the base and the slant height of the pyramid. Suppose that a regular pyramid has a square base whose side is s and whose slant height is /.

1. Where does the vertex of the pyramid lie?

Assume that a plane perpendicular to the base is passed through the vertex of the pyramid so that the plane intersects the midpoints of two opposite sides of the square.

2. Draw a picture of the intersection of this plane with the pyramid.

3. What type of fi gure is formed?

4. Draw the altitude of this triangle, and label it h.In terms of the pyramid, what is h?

5. What theorem expresses h in terms of / and s?

6. Express h in terms of / and s.

7. What is the area of the base of the pyramid?


Suppose that the base of the pyramid is a regular n-gon with a side of length s. Pass a plane through the vertex of the pyramid perpendicular to the base so that it intersects the midpoint of a side of the pyramid.

9. Draw the intersection. Let V denote the vertex of the pyramid, C the center of the base, and M the midpoint of a side of the polygon.

10. What does VM represent?

11. What does CM represent in terms of the polygon?

12. What does VC represent in terms of the pyramid?

13. Let VC 5 h and CM 5 a. Compute h in terms of a and /.

14. Find the area of the base in terms of n, s, and a.


on a line perpendicular to the base at the center of the square

isosceles triangle

the height of the pyramid

Pythagorean Theorem

slant height

apothem

height

h 5"<2 2 a2

A 5 nas2

nas6 "<2 2 a2

s2

s2

6 "4<2 2 s2

h 5 12 "4<2 2 s2

2., 4.

h,

s2

9., 13.

h,

M a C

V


49

Name Class Date

11-5 Reteaching Volumes of Pyramids and Cones

Problem

What is the volume of the square pyramid?

Sometimes the height of a triangular face in a square pyramid is not given. Here the slant height and the lengths of the sides of the base are given. Use what you know about right triangles to fi nd the missing value. Th en calculate the volume as usual.

72 1 x2 5 252 Use the Pythagorean Theorem.

49 1 x2 5 625 Substitute.

x2 5 625 2 49 Isolate the variable.

x2 5 576 Simplify.

x 5 24 cm Find the square root of each side.

Volume of the pyramid:

V 5 13 Bh Use the formula for volume of a pyramid.

513(14 3 14)(24) Substitute.

5 1568 cm3 Simplify.

Th e volume of the square pyramid is 1568 cm3.

Exercises

Find the volume of each pyramid. Round to the nearest whole number.

1. 2. 3.

4. 5. 6.

25 cm

14 cm

14 cm

45 cm

54 cm

54 cm 32 in.32 in.

34 in.

150 m2

3 m

13 in.

10 in.10 in.

36 yd

400 yd2

18 cm

8 cm2

34,992 cm3 10,240 in.3 150 m3

400 in.3 4800 yd3 48 cm3


50

Name Class Date



Problem

What is the volume of the cone?

Find the height of the cone.

132 5 h2 1 52 Use the Pythagorean Theorem.

169 5 h2 1 25 Substitute.

h2 5 144 Simplify.

h 5 12 Take the square root of each side.

Find the volume of the cone.

V 5 13pr

2h Use the formula for the volume of a cone.

513p(5)2 ? 12 Substitute.

5 100p Simplify.

< 314.2

Th e volume of the cone is about 314.2 cm2.

Exercises

7. From the fi gures shown below, choose the pyramid with volume closest to the volume of the cone at the right.

A. B. C.

Find the volume of each fi gure. Round your answers to the nearest tenth.

8. 9. 10. 11.

h 13 cm

5 cm

2.5 in.

7 in.

5 in.

5 in.

7 in.

3 in.

15 in.

3 in.

2 in.8 in.

8 in.

10 cm

6 cm

h

17 m15 cm

2 ft

6 ft

12.7 mh

4.1 m

1005.3 m3

18.8 ft3

B

301.6 cm3211.6 m3


51

Name Class Date

11-6 Additional Vocabulary Support Surface Areas and Volumes of Spheres

For Exercises 1–6, match the term in Column A with its defi nition in Column B. Th e fi rst one is done for you.

Column A Column B

sphere a segment with both endpoints on a sphere that passes through the center of the sphere

1. center of a sphere one of the two parts of a sphere on either side of a great circle

2. radius of a sphere the set of all points in space that are a given distance from a given point

3. diameter of a sphere a segment that has one endpoint at the center of a sphere and the other endpoint on the sphere

4. great circle a point that is the same distance from all points on a sphere

5. circumference of a sphere the circumference of any great circle of a sphere

6. hemisphere the intersection of a sphere and the plane containing the center of a sphere

For Exercises 7–10, match the term in Column A with its formula in Column B.

Column A Column B

7. circumference of a sphere V 5 43pr

3

8. surface area of a sphere C 5 2pr

9. volume of a sphere S.A. 5 4pr2

10. diameter of a sphere D 5 2r


52

Name Class Date

11-6 Think About a Plan Surface Areas and Volumes of Spheres

Meteorology On September 3, 1970, a hailstone with diameter 5.6 in. fell at Coff eyville, Kansas. It weighed about 0.018 lb/in.3 compared to the normal

0.033 lb/in.3 for ice. About how heavy was this Kansas hailstone?


1. What is the situation described in the problem?

2. What key piece of information is unstated or implicit?

3. What question do you have to answer?

4. Is there any unnecessary information? Explain.


5. What formula can you use to fi nd the volume of the hailstone?

6. How can you use the volume of the hailstone to fi nd its weight?

Getting an Answer

7. What is the volume of the hailstone? Show your work.

8. How do you fi nd the weight of the hailstone?

A hailstone fell from the sky.

The hailstone is spherical.

How heavy was the hailstone?

Yes; we don’t need to know the normal lb/in.3 of ice.

You can use the formula V 5 43 πr

3.

Multiply the volume by the density, 0.018 lb/in.3.

V 5 43 π(5.6

2 )3 N 92 in.3

The weight is 92 3 0.018 5 1.656 lb or 1 lb 10.5 oz.


53

Name Class Date


Surface Areas and Volumes of Spheres

Find the surface area of the sphere with the given diameter or radius. Leave your answer in terms of π.

1. d 5 8 ft 2. r 5 10 cm

3. d 5 14 in. 4. r 5 3 yd

Find the surface area of each sphere. Leave each answer in terms of π.

5. 6. 7.

8. 9. 10.

Use the given circumference to fi nd the surface area of each spherical object. Round your answer to the nearest whole number.

11. an asteroid with C 5 83.92 m 12. a meteorite with C 5 26.062 yd

13. a rock with C 5 16.328 ft 14. an orange with C 5 50.24 mm

Find the volume of each sphere. Give each answer in terms of π and rounded to the nearest cubic unit.

15. 16. 17.

18. 19. 20.

A sphere has the volume given. Find its surface area to the nearest whole number.

21. V 5 1200 ft3 22. V 5 750 m3 23. V 5 4500 cm3

5 cm 4 yd 6 ft

9 in.20 mm

9 m

18 mm 5 yd 3 m

20 in.

S.A. 16 cm2 S.A. 64 cm2

64π ft2

36π ft2

196π in.2

324π in.2

972π mm3; 3054 mm3

1333 13 π in.3; 4189 in.3

100π cm2

400π cm2

400π mm2

166 23 π yd3; 524 yd3

10 23 π cm3; 34 cm3

2242 m2 216 yd2

803 mm285 ft2

81π m2

36π m3; 113 m3

85 13 π cm3; 268 cm3

36π yd2

64π yd2

546 ft2 399 m2 1318 cm2


54

Name Class Date

Find the volume in terms of π of each sphere with the given surface area.

24. 900p in.2 25. 81p in.2 26. 6084p m2

27. Th e diff erence between drizzle and rain has to do with the size of the drops, not how much water is actually falling from the sky. If rain consists of drops larger than 0.02 in. in diameter, and drizzle consists of drops less than 0.02 in. in diameter, what can you say about the surface area and volume of rain and drizzle?

28. A spherical scoop of ice cream with a diameter of 4 cm rests on top of a sugar cone that is 10 cm deep and has a diameter of 4 cm. If all of the ice cream melts into the cone, what percent of the cone will be fi lled?

29. Point A is the center of the sphere. Point C is on the surface of the sphere. Point B is the center of the circle that lies in plane P and includes point C. Th e radius of the circle is 12 mm. AB 5 5 mm. What is the volume of the sphere to the nearest cubic mm?

30. Writing What are the formulas for the volumes of a sphere, a cone with a height equal to its radius, and a cylinder with its height equal to its radius? How are these formulas related?

31. Candlepin bowling balls have no holes in them and are smaller than the bowling balls used in tenpin bowling. Th e regulation size is 4.5 in. in diameter, and their density is 0.05 lb/in.3. What is the regulation weight of a candlepin bowling ball? Round your answer to the nearest tenth of a pound.

32. Find the radius of a sphere such that the ratio of the surface area in square inches to the volume in cubic inches is 4 i1.

33. Find the radius of a sphere such that the ratio of the surface area in square feet to the volume in cubic feet is 2 i5.

PAC B



4500π in.3 121.5π in.3 79,092π m3

rain: S.A. S 0.0013 in.2, V S 0.000004 in.3; drizzle: S.A. R 0.0013 in.2, V R 0.000004 in.3

80%

9203 mm2

2.4 lb

0.75 in.

7.5 ft or 7 ft 6 in.

Sphere: V 5 43 πr

3; cone: V 5 13 πr

3; cylinder: V 5 πr 3; the volume of a sphere is equal

to the sum of the volume of a cone and a cylinder with height equal to their radii.


55

Name Class Date



Find the surface area of the sphere with the given diameter or radius. Leave your answer in terms of π.

1. r 5 6 ft 2. d 5 10 cm

3. r 5 8 in. 4. d 5 4 yd

Find the surface area of each sphere. Leave each answer in terms of π.

5. To start, use the formula for the surface area of a sphere. Th en determine the radius and substitute it into the formula.

S.A. 5 4pr2 5 4p ?u2

6. 7. 8.

Use the given circumference to fi nd the surface area of each spherical object. Round your answer to the nearest tenth.

9. a baseball with C 5 9.25 in. 10. a softball with C 5 28.25 cm

11. a basketball with C 5 2.98 ft 12. a bowling ball with C 5 26.7 in.

Find the volume of each sphere. Give each answer in terms of π and rounded to the nearest cubic unit.

13. To start, use the formula for the surface area of a sphere. Th en determine the radius, and substitute it into the formula.

V 5 43pr3 5

43p ?u

3

14. 15. 16.

3 cm

7 yd25 in. 30 mm

6 mm

9 m 15 in. 36 cm

288π mm3; 905 mm3

972π m3; 3054 m3562

12π in.3; 1767 in.3 7776π cm3; 24,429 cm3

3

6

100π cm2

36π cm2

16π yd2

196π yd2

144π ft2

256π in.2

27.2 in.2

2.8 ft2

254.0 cm2

226.9 in.2

625π in.2 900π mm2


56

Name Class Date


17. V 5 14,130 ft3 18. V 5 4443 m3

19. V 5 100 in.3 20. V 5 31,400 mi3

21. A spherical scoop of ice cream with a diameter of 5 cm rests on top of a sugar cone that is 12 cm deep and has a diameter of 5 cm. If all of the ice cream melts into the cone, what percent of the cone will be fi lled? Round to the nearest percent.

22. Writing A cylinder, a cone, and a sphere have the dimensions indicated in the diagram below.

a. What are the formulas for the volume of the cone and the volume of the cylinder in terms of r? Give each answer in terms of p.

b. If r 5 9 in., what are the volumes of the cone, cylinder, and sphere?

c. How are the volumes related?

d. How can you show that this relationship is true for all values of r?

23. A bowling ball must have a diameter of 8.5 in. If the bowling ball weighs 16 lb, fi nd the density (lb/in.3) of the bowling ball. Density is the quotient of weight divided by volume. Round your answer to the nearest hundredth.

24. Open-Ended Draw two spheres such that the volume of one sphere is eight times the volume of the other sphere.

5 cm

12 cm

r r

rr

r



243π, 729π, 972π

2826 ft2 1307 m2

104 in.2 4813 mi2

83%

The volume of a sphere is equal to the sum of the volumes of the cone and the cylinder.

by using the formulas for the volumes to show that 13 πr3 1 πr3 5 43 πr3

0.05 lb/in.3

Answers may vary. The larger sphere’s radius will be twice the radius of the smaller sphere.

cone: V 5 13 πr3; cylinder: V 5 πr3


57

Name Class Date

11-6 Standardized Test Prep Surface Areas and Volumes of Spheres

Multiple Choice


1. What is the approximate volume of the sphere?

524 m3 1256 m3

1000 m3 1570 m3

2. What is the approximate surface area of the sphere?

225 yd2 1767 yd2

707 yd2 5301 yd2

3. What is the approximate volume of the sphere if the surface area is 482.8 mm2?

998 mm3 1126 mm3 2042 mm3 2993 mm3

4. What is the approximate surface area of the sphere?

342.3 km2 903.4 km2

451.9 km2 2713 km2

5. What is the approximate radius of a sphere whose volume is 1349 cm3?

5.7 cm 6.9 cm 11 cm 14.7 cm

Short Response

6. Suppose a wealthy entrepreneur commissions the design of a spherical spaceship to house a small group for a week in orbit around the Earth. Th e designer allocates 1000 ft3 for each person, plus an additional 4073.5 ft3 for various necessary machines. As in a recreational vehicle, the personal space is largely occupied by items such as beds, shower and toilet facilities, and a kitchenette. Th e diameter of the ship is 26.8 ft. What is the volume of the spaceship, and for approximately how many people is the ship designed?

10 m

15 yd

C 37.68 km

A

G

A

B

[2] V 5 10,078.7 ft3; 6 people [1] incorrect volume or incorrect number of people [0] no response or incorrect response

G


58

Name Class Date

11-6 Enrichment Surface Areas and Volumes of Spheres

Two spheres of radius R and r are concentric if the centers of both spheres are the same point. Assume that r S R.

1. What is the volume of the outer (larger) sphere?

2. What is the volume of the inner (smaller) sphere?

3. What is the volume of the space between the smaller and the larger sphere?

Th e radius of Earth is approximately 4000 mi, and the radius of its core, which is a sphere of molten metals, is about 800 mi.

4. What is the volume of Earth that lies outside the core to the nearest billion mi3?

5. Use (x3 2 y3) 5 (x 2 y)(x2 1 xy 1 y2) to factor the expression for the volume lying between two concentric spheres.

Th e thickness d of the thin spherical shell representing the diff erence between the two spheres is defi ned to be d 5 r 2 R.

6. Using r < R, compute the approximate volume of a thin spherical shell of radius R and thickness d.

7. To see the accuracy of this approximation, fi ll in the table to two decimal places.

Most golf balls consist of three concentric spheres: a liquid-fi lled center, elastic windings, and a dimpled cover. Th e size and materials of these three components vary for each type of golf ball. A specifi c type of golf ball has radius 0.84375 in. Th e center of the golf ball has radius 0.565 in., and the cover has thickness 0.05 in.

8. What is the volume of the outside cover to the nearest thousandth?

9. What is the volume of the elastic winding to the nearest hundreth?

R d Approximate Volume of ShellExact Volume of Shell

10

50

100

0.1

1

3

elastic windings

liquid-filled center

dimpled cover

43 πr

3

43 πR

3

43 π(r

3 2 R 3)

266,000,000,000 mi3

43 π(r 2 R)(r

2 1 rR 1 R2)

43 πd(3R2) 5 4πdR2

0.421 in.3

1.34 in.3

126.92

32,048.43

388,413.95

125.66

31,415.93

376,991.12


59

Name Class Date

11-6 Reteaching Surface Areas and Volumes of Spheres

Problem

What are the surface area and volume of the sphere?

Substitute r 5 5 into each formula, and simplify.

S.A. 5 4pr 2 V 5 4

3pr 3

5 4p(5)2 543p(5)3

5 100p 5500p

3

< 314.2 < 523.6

Th e surface area of the sphere is about 314.2 in.2. Th e volume of the sphere is about 523.6 in.3.

Exercises

Use the fi gures at the right to guide you in completing the following.

1. Use a compass to draw two circles, each with radius 3 in. Cut out each circle.

2. Fold one circle in half three successive times. Number the central angles 1 through 8.

3. Cut out the sectors, and tape them together as shown.

4. Take the other circle, fold it in half, and tape it to the rearranged circle so that they form a quadrant of a sphere.

5. Th e area of one circle has covered one quadrant of a sphere. How many circles would cover the entire sphere?

6. How is the radius of the sphere related to the radius of the circle?

Find the volume and surface area of a sphere with the given radius or diameter. Round your answers to the nearest tenth.

7. 8. 9.

5 in.

6

2

84

3

57

1

6

8

2

435

7

1

13

57 8

6

42

10 in. 3 cm 24 m

V 5 523.6 in.3; S.A. 5 314.2 in.2

V 5 113.1 cm3; S.A. 5 113.1 cm2

V 5 7,238.2 m3; S.A. 5 1809.6 m2





four circles

They are the same.


60

Name Class Date



Find the volume and surface area of the sphere. Round to the nearest tenth.

10. 11. 12.

13. 14. 15.


16. 1436.8 mi3 17. 808 cm3 18. 72 m3

Find the volume of each sphere with the given surface area. Round to the nearest whole number.

19. 435 yd2 20. 907 cm2 21. 28 m2

22. Visualization Th e region enclosed by the semicircle at the right is revolved completely about the x-axis.

a. Describe the solid of revolution that is formed.

b. Find its volume in terms of p.

c. Find its surface area in terms of p.

23. Th e sphere at the right fi ts snugly inside a cube with 18 cm edges. What is the volume of the sphere? What is the surface area of the sphere? Leave your answers in terms of p.

14 in. 700 m

2 cm

10 m

2 ft

7 m

24 2 4

2

4

y

xO

18 cm

616 mi2 420 cm2 84 m2

853 yd3 2569 cm3 14 m3

a sphere

85 13 π units3

64π units2

972π cm3; 324π cm2

S.A. 5 2463.0 in.2; V 5 11,494.0 in.3

S.A. 5 1256.6 m2;V 5 4188.8 m3

S.A. 5 6,157,521.6 m2; V 5 1,436,755,040.2 m3

S.A. 5 50.3 ft2;V 5 33.5 ft3

S.A. 5 12.6 cm2; V 5 4.2 cm3

S.A. 5 153.9 m2;V 5 179.6 m3


61

Name Class Date

11-7 Additional Vocabulary SupportAreas and Volumes of Similar Solids

Choose the word from the list below that best matches each description.

area lateral area linear dimensions

proportion ratio scale factor

similar surface area volume

1. a comparison of two quantities

2. fi gures that are the same shape and have corresponding sides that are proportional

3. equal ratios

4. the ratio of the corresponding dimensions of similar fi gures

Use a word from the list above to complete each sentence.

5. Th e sum of the areas of each face of a solid fi gure that is not a base is the .

6. To fi nd the of a rectangular prism, multiply the area of the base by the height.

7. A solid’s length, width, and height are its .

8. Th e sum of the areas of each face of a solid fi gure is its .

Circle the correct value for the fi gures at the right.

9. surface area of fi gure A 400 ft2 448 ft2

10. volume of fi gure B 60 ft3 112 ft3

11. scale factor of fi gure A 2 i1 3 i1 10 i1to fi gure B

Multiple Choice

12. If the scale factor between two fi gures is 2 i1, what is the ratio of the volume of the fi gures?

1 i3 3 i1 4 i1 8 i1

20 ft

6 ft

3 ft2 ft

10 ft

4 ft

A B

ratio

lateral area

volume

linear dimensions

surface area

similar

proportion

scale factor

D


62

Name Class Date

11-7 Think About a Plan Areas and Volumes of Similar Solids

Reasoning A carpenter is making a blanket chest based on an antique chest. Both chests have the shape of a rectangular prism. Th e length, width, and height of the new chest will all be 4 in. greater than the respective dimensions of the antique. Will the chests be similar? Explain.


1. How much longer will the blanket chest be than the antique chest? How much wider will it be? How much taller will it be? Draw a diagram of the blanket chest showing the increase in each dimension.

2. If two solids are similar, what must be true about their corresponding dimensions?


3. Practice explaining your reasoning. Imagine a chest has dimensions 3 ft-by-3 ft-by-3 ft. Add 4 in. to each dimension. Is the new chest similar to the old chest? Explain.

4. Practice explaining your reasoning. Suppose a chest has dimensions 4 ft-by-3 ft-by-3 ft. Add 4 in. to each dimension. Is the new chest similar to the old chest? Explain.

Getting an Answer

5. How can you generalize the dimensions of the old chest?

6. Using the dimensions of the chest, determine whether the new chest and the old chest will be similar.

The new chest will be 4 in. longer, wider, and taller than the antique chest.

Each pair of corresponding dimensions must satisfy the same ratio.

Yes; the ratios between each pair of corresponding dimensions are equal.

No; the new chest is not similar because the ratios between each pair of

corresponding dimensions are not equal.

Sample: The dimensions can be represented by the variables <, w, and h.

Unless < 5 w 5 h, then the ratios between each pair of corresponding

dimensions will not be equal, and they will not be similar.


63

Name Class Date


Areas and Volumes of Similar Solids

Are the two fi gures similar? If so, give the scale factor of the fi rst fi gure to the second fi gure.

1. 2.

3. 4.

5. two cubes, one with 5-in. edges, the other with 6-in. edges

6. a cylinder and a cone, each with 6-m radii and 4-m heights

Each pair of fi gures is similar. Use the given information to fi nd the scale factor of the smaller fi gure to the larger fi gure.

7. 8.

9. 10.

Th e surface areas of two similar fi gures are given. Th e volume of the larger fi gure is given. Find the volume of the smaller fi gure.

11. S.A. 5 36 m2 12. S.A. 5 108 in. 2 13. S.A. 5 49 m2

S.A. 5 225 m2 S.A. 5 192 in.2 S.A. 5 441 m2

V 5 750 m3 V 5 1408 in.3 V 5 432 m3

14. A shipping box holds 350 golf balls. A larger shipping box has dimensions triple the size of the other box. How many golf balls does the larger box hold?

15 in.

10 in.

6 in.

4 in. 3 cm3 cm

3 cm 5 cm

5 cm5 cm

4 ft

6 ft

10 ft

9 ft 12 m

6 m

9 m

2 m

V 64 in.3 V 125 in.3 V 216 cm3V 125 cm3

S.A. 150 m2 S.A. 294 m2S.A. 121 ft2S.A. 36 ft2

no

yes; 5 i6

no

48 m3 594 in.3 16 m3

9450

4 i5 5 i6

5 i7 6 i11

yes; 5 i2 yes; 3 i5

no


64

Name Class Date

Th e volumes of two similar fi gures are given. Th e surface area of the smaller fi gure is given. Find the surface area of the larger fi gure.

15. V 5 8 m3 16. V 5 125 in.3 17. V 5 3 ft3

V 5 27 m3 V 5 216 in.3 V 5 375 ft3

S.A. 36 5 m2 S.A. 5 200 in.2 S.A. 5 4 ft2

18. A cylindrical thermos has a radius of 2 in. and is 5 in. high. It holds 10 fl oz. To the nearest ounce, how many ounces will a similar thermos with a radius of 3 in. hold?

19. Compare and Contrast You have a set of three similar nesting gift boxes. Each box is a regular hexagonal prism. Th e large box has 10-cm base edges. Th e medium box has 6-cm base edges. Th e small box has 3-cm base edges. How does the volume of each box compare to every other box?

20. Two similar pyramids have heights 6 m and 9 m.

a. What is their scale factor?

b. What is the ratio of their surface areas?

c. What is the ratio of their volumes?

21. A small, spherical hamster ball has a diameter of 8 in. and a volume of about 268 in.3. A larger ball has a diameter of 14 in. Estimate the volume of the larger hamster ball.

22. Error Analysis A classmate says that a rectangular prism that is 6 cm long, 8 cm wide, and 15 cm high is similar to a rectangular prism that is 12 cm long, 14 cm wide, and 21 cm high. Explain your classmate’s error.

23. Th e lateral area of two similar cylinders is 64 m2 and 144 m2. Th e volume of the larger cylinder is 216 m3. What is the volume of the smaller cylinder?

24. Th e volumes of two similar prisms are 135 ft3 and 5000 ft3.

a. Find the ratio of their heights.

b. Find the ratio of the area of their bases.



81 m2 288 in.2 100 ft2

34 fl oz

2 i3

4 i9

8 i27

3 i10

9 i100

1436 in.3

64 m3

The ratios of the dimensions are not the same: the ratio of the lengths is 1 i2, the ratio of the widths is 4 i7, and the ratio of the heights is 5 i7. Each dimension of the larger prism is 6 cm greater.

The ratio of the volumes of the large box to the medium box is 125 i27; the ratio of the volumes of the large box to the small box is 1000 i27; the ratio of the volumes of the medium box to the small box is 8 i1.


65

Name Class Date



Are the two fi gures similar? If so, give the scale factor of the fi rst fi gure to the second fi gure.

1. 2.

3. 4.

5. two cubes, one with 6-in. edges, the other with 8-in. edges

6. a cylinder and a cone, each with 9-m radii and 5-m heights


7. To start, write a proportion using the ratio of the volumes of the solids.

a3

b3 564

216

8. 9.

10. Two similar cones have heights 4 m and 12 m. a. What is their scale factor? b. What is the ratio of their surface areas? c. What is the ratio of their volumes?

11. A shipping box holds 450 golf balls. A larger shipping box has dimensions triple the size of the other box. How many golf balls does the larger box hold?

6 in.4 in.2 in. 3 in. 6 in.

9 in.

4 cm 5 cm

8 cm10 cm

4 in.4 in.

4 in.6 in.

6 in.8 in.

3 cm4 cm

V 64 cm3 V 216 cm3

V 128 mm3 V 686 mm3 S.A. 54 m2 S.A. 150 m2

yes; 4 i5

yes; 3 i4yes; 4 i3

yes; 2 i3

2 i3

4 i7

1 i31 i9

1 i27

12,150 balls

3 i5

no

no


66

Name Class Date


12. S.A. 5 94 m2

S.A. 5 846 m2

V 5 1620 m3

To start, fi nd the scale factor a ib.

a2

b2 5uu

13. S.A. 5 240 m2

S.A. 5 1500 m2

V 5 1562.5 m3

14. S.A. 5 96 in.2

S.A. 5 216 in.2

V 5 216 in.3

Th e volumes of two similar fi gures are given. Th e surface area of the larger fi gure is given. Find the surface area of the smaller fi gure.

15. V 5 384 m3

V 5 10,368 m3

S.A. 5 3168 m2

16. V 5 216 in.3

V 5 1728 in.3

S.A. 5 864 in.2

17. A cylindrical thermos has a radius of 3 in. and is 12 in. high. It holds 20 fl oz. To the nearest ounce, how many ounces will a similar thermos with a radius of 4 in. hold?

18. You have a set of three similar gift boxes. Each box is a rectangular prism. Th e large box has 15-cm base edges. Th e medium box has 10-cm base edges. Th e small box has 5-cm base edges. How does the volume of each box compare to every other box?

19. A baseball and a softball are similar in shape. Th e baseball has a radius of 1.25 in. and a volume of 8.18 in.3. If the volume of a softball is 65.44 in.3, what is the radius of the softball?

20. Error Analysis A classmate says that a rectangular prism that is 9 cm long, 12 cm wide, and 15 cm high is similar to a rectangular prism that is 12 cm long, 16 cm wide, and 21 cm high. Explain your classmate’s error.

21. Th e volumes of two similar prisms are 512 ft3 and 8000 ft3.

a. Find the ratio of their heights.

b. Find the ratio of the area of their bases.



60 m3

94

846

100 m3 64 in.3

352 m2 216 in.2

47 fl oz

The ratio of the volume of the large box to that of the medium box is 27 i8; the ratio of the volume of the large box to that of the small box is 27 i1; the ratio of the volume of the medium box to that of the small box is 8 i1.

The ratios of the dimensions are not the same. The ratio of the lengths is 3 i4, the ratio of the widths is 3 i4, but the ratio of the heights is 5 i7.

2 i5

2.5 in.

4 i25


67

Name Class Date

11-7 Standardized Test Prep Areas and Volumes of Similar Solids

Multiple Choice


1. Which of the fi gures shown below are similar?

2. Th e measure of the side of a cube is 6 ft. Th e measure of the side of a second cube is 18 ft. What is the scale factor of the cubes?

1 i2 1 i3 1 i9 1 i27

3. What is the ratio of the surface areas of the similar square pyramids at the right?

4 i5 16 i25

8 i10 64 i125

4. What is the ratio of the volumes of the similar square pyramids above?

4 i5 8 i10 16 i25 64 i125

5. Th e surface areas of two similar triangular prisms are 132 m2 and 297 m2. Th e volume of the smaller prism is 264 m3. What is the volume of the larger prism?

594 m3 891 m3 1336.5 m3 3007.125 m3

Short Response

6. A medium-sized box can hold 55 T-shirts. If the dimensions of a jumbo box are three times that of the medium box, how many T-shirts can the jumbo box hold? Explain.

4 m

6 m

3m

8 m4 cm

7 cm

3 cm

3 cm

6 cm

2 cm

4 in.

8 in.3 in.12 in. 20 m

10 m8 m

4 m

10 ft

15 ft

8 ft

12 ft

D

G

C

I

B

[2] 1485 T-shirts; if the dimensions of the jumbo box are three times those of the medium box, the scale factor is 1 i3. The ratio of the volumes is 13 i33, or 1 i27. Set up a proportion to solve: 1

27 555x ; x 5 27 ? 55 5 1485. [1] error in calculation,

or work not shown [0] incorrect or no response


68

Name Class Date

11-7 Enrichment Areas and Volumes of Similar Solids

Use each situation to complete the exercises that follow. Any costs given are proportional to the number of units.

Two storage bins are built in the form of rectangular prisms, and the two bins are similar. One stores wheat at a cost of $0.15 per bushel, and the other stores corn at a cost of $0.20 per bushel. Th e bin storing the wheat has a square base 80 ft on a side and is 120 ft tall.

1. If the cost of storing the wheat is $8000, and the cost of storing the corn is $36,000, fi nd the height and the length to the nearest whole number of a side of the base of the bin storing corn.

A sphere with a 4-in. radius is being plated with silver for a cost of $30.

2. If gold plating costs 75% more than silver plating, how much will it cost to gold-plate a sphere with radius 6 in.?

3. To the nearest hundredth, what is the radius of a sphere that is plated with silver for a cost of $80?

4. To the nearest hundredth, what is the radius of a sphere that is plated with gold for a cost of $100?

Two similar cylinders contain juice. Th e fi rst cylinder has radius 6 in. and height 10 in., contains orange juice, and sells for $2.40.

5. If grapefruit juice costs two-thirds of the price of an equal volume of orange juice, what is the cost of a container of grapefruit juice that has radius 9 in. and height 15 in.?

6. To the nearest hundredth, what is the radius of a similar container holding $4.00 worth of grapefruit juice?

7. To the nearest hundredth, what is its height?

Two similar cones have a combined volume of 400 in.3, and the larger cone holds

80 in.3 more than the smaller cone, which has a radius of 3 in.

8. To the nearest hundredth, what is the height of the smaller cone?

9. What is the radius of the larger cone?

10. What is the height of the larger cone?

180 ft; 120 ft

$118.13

6.53 in.

5.52 in.

$5.40

8.14 in.

13.57 in.

16.98 in.

3.43 in.

19.43 in.


69

Name Class Date

11-7 Reteaching Areas and Volumes of Similar Solids

When two solids are similar, their corresponding dimensions are proportional.

Rectangular prisms A and B are similar because the ratio of their corresponding dimensions is 23.

height: 8 m12 m 5

23 length: 2 m

3 m 523 width: 4 m

6 m 523

Th e ratio of the corresponding dimensions of similar solids is called the scale factor. All the linear dimensions (length, width, and height) of a solid must have the same scale factor for the solids to be similar.

Areas and Volumes of Similar SolidsArea Volume• Th e ratio of corresponding areas of similar

solids is the square of the scale factor.• Th e ratio of the volumes of similar

solids is the cube of the scale factor.

• Th e ratio of the areas of prisms A and B is 2

2

32, or 49.• Th e ratio of the volumes of prisms

A and B is 23

33, or 827.

Problem

Th e pyramids shown are similar, and they have volumes of 216 in.3 and 125 in.3 Th e larger pyramid has surface area 250 in.2

What is the ratio of their surface areas? What is the surface area of the smaller pyramid?

By Th eorem 11-12, if similar solids have similarity ratio a ib, then the ratio of their volumes is a3 ib3. So,

a3

b3 5216125

ab 5

65 Take the cube root of both sides to get a ib.

a2

b2 53625 Square both sides to get a2 ib2.

Ratio of surface areas 5 36 i25

If the larger pyramid has surface area 250 in.2, let the smaller pyramid have surface area x. Th en,

250x 5

3625

36x 5 6250

x < 173.6 in.2

Th e surface area of the smaller pyramid is about 173.6 in2.

3 m6 m

12 m

2 m4 m

8 m


70

Name Class Date



ExercisesFind the scale factors.

1. Similar cylinders have volumes of 200p in.3 and 25p in.3

2. Similar cylinders have surface areas of 45p in.2 and 20p in.2

Are the two fi gures similar? If so, give the scale factor.

3. 4.


5. 6.

Find the ratio of volumes.

7. Two cubes have sides of length 4 cm and 5 cm.

8. Two cubes have surface areas of 64 in.2 and 49 in.2


9. S.A. 5 16 cm2 10. S.A. 5 6 ft2 11. S.A. 5 45 m2

S.A. 5 100 cm2 S.A. 5 294 ft2 S.A. 5 80 m2

V 5 500 cm3 V 5 3430 ft3 V 5 320 m3

Th e volumes of two similar fi gures are given. Th e surface area of the smaller fi gure is given. Find the surface area of the larger fi gure.

12. V 5 12 in.3 13. V 5 6 cm3 14. V 5 40 ft3

V 5 96 in.3 V 5 384 cm3 V 5 135 ft3

S.A. 5 12 in.2 S.A. 5 6 cm2 S.A. 5 20 ft2

18 m

8 m9 m

4 m

10 in.

4 in.5 in.

2 in.

6 in.

3 in.

V 135 in.3 V 320 in.3 S.A. 32 cm2 S.A. 162 cm2

yes; 1 i2 no

3 i4 4 i9

64 i125

512 i343

32 cm3 10 ft3 135 m3

48 in.2 96 cm2 45 ft2

2 i 1

3 i 2


71

Name Class Date

Chapter 11 Quiz 1 Form G

Lessons 11-1 through 11-3

Do you know HOW?

1. How many faces, edges, and vertices are in the solid? List them.

2. What is a net for the solid in Exercise 1? Verify Euler’s Formula for the net.

3. Find the surface area of the prism.

5

43

4. Find the surface area of the cylinder in terms of p.

6

6

5. Find the lateral area of the cone in terms of p.

10 86

6. Find the surface area of the cone in terms of p.

10 68

Do you UNDERSTAND?

7. Vocabulary Suppose you build a polyhedron from two hexagons and six rectangles. Without using Euler’s Formula, how many edges does the solid have? Explain.

8. Error Analysis Your friend draws a net of a prism. What is your friend’s error?

9. Compare and Contrast How are the formula for the surface area of a pyramid and the surface area of a cone alike? How are they diff erent?

10. Error Analysis A cone has radius 3 and height 4. Your classmate calculates its surface area as shown: S.A. 5 p(5)2 1 p(3)2 5 25p 1 9p 5 3p. What is the error? Explain.

D

AB

C

4 6 9 1

faces: 4; edges: 6; vertices: 4; faces: kABC, kABD, kACD, kBCD; edges: AB, AC, AD, BC, BD, CD; vertices: A, B, C, D

18 edges, 6 on each base and 6 more connecting the bases: 6 1 6 1 6 5 18.

94

Sample:

54π

60π 144π

The net is missing one of the bases.

Answers may vary. Sample: They are both of the form S.A. 5 L.A. 1 B; the formulas for the lateral areas are different.

The student uses L.A. 5 π<2 instead of L.A. 5 πr <.


72

Name Class Date

Do you know HOW?

What is the volume of each fi gure? If necessary, round to the nearest tenth.

1. 2.

3. 4.

5. Th e surface area of a sphere is 100p. What is its volume in terms of p?

6. Th e surface areas of two similar containers are 1125 cm2 and 375 cm2. Th e volume of the smaller container is 450 cm3. What is the volume of the larger container, to the nearest whole number?

Do you UNDERSTAND?

7. Reasoning A cylinder has a height of 4 mm and a volume of 64 mm3. What is the radius of the base? If necessary, round to the nearest tenth.

8. Error Analysis Two cones have the same volume, but diff erent heights. Th e taller cone is four times taller than the shorter cone. Your friend concludes that the radius of the shorter cone must be four times longer than the radius of the taller cone. What is his error?

9. Error Analysis Th e ratio of the surface areas of two solids is 9 i1 and the ratio of their two volumes is also 9 i1. Your classmate concludes that the two solids are similar, with a scale factor of 9 i1. Explain and correct your classmate’s error.

4 cm3 cm

5 cm

6 cm

3 in.

5 in.

3 mm

7 mm

3 mm

9 m

5 m

Chapter 11 Quiz 2 Form G


2338 cm3

2.3 mm

5003 π

36 cm3

21 mm3

141.4 in.3

235.6 m3

Answers will vary. Sample: The friend has not considered that the area of the base is B 5 πr2. So, the radius of the shorter cone is only twice as long as the radius of the taller cone.

Answers may vary. Sample: If the ratio of the surface areas of two similar solids is 9 5 3 ? 3, then the ratio of their volumes is 3 ? 3 ? 3 5 27. Therefore, two solids whose surfaces and volumes both have a ratio of 9 cannot be similar.


73

Name Class Date

Chapter 11 Test Form G

Surface Area and Volume

Do you know HOW?

Draw a net for each fi gure. Label the net with appropriate dimensions.

1. 2.

Use the polyhedron at the right for Exercises 3 and 4.

3. Verify Euler’s formula for the fi gure.

4. Draw a net for the polyhedron. Verify F 1 V 5 E 1 1 for the net.

5. What is the number of edges in a prism with hexagonal bases?

Describe the cross section formed in each diagram.

6. 7.

8. A food company makes regular and tall soup cans. Th e area of the base of both cans is 30 cm2. Th e volume of the regular can is 270 cm3. Th e tall can is 2 cm taller. What is the volume of the tall soup can?

Find the surface area and volume of each fi gure to the nearest tenth.

9. 10.

5 cm

4 cm6 cm3 cm

2 in.

5 in.

3 in.

2 in.

2 in.

7 in.

10 cm

9 cm

6 1 6 5 10 1 2

18

circle

isosceles triangle

330 cm3

S.A. 5 64 in.2; V 5 28 in.3 S.A. 5 439.8 cm2; V 5 706.9 cm3

6 10 15 1

6 cm

3 cm 3 cm

4 cm 4 cm

5 cm

5 cm 5 cm

5 in.

2 in.

2 in.2 in.

3 in.

3 in. 3 in.

2 in.


74

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Find the surface area and volume of each fi gure. Round to the nearest tenth.

11. 12.

13. 14.

15. Th e circumference of a standard baseball is about 9 in. About how many square in. of horsehide are required to cover 100 balls, to the nearest whole number?

16. Th e volumes of two similar prisms are 891 cm3 and 33 cm3. Th e surface area of the larger prism is 153 cm2. What is the surface area of the smaller prism?

Do you UNDERSTAND?

17. Reasoning Every cross section parallel to the base of a solid is a circle. If all those circles are congruent, what type of solid is it? If none of those circles are congruent, what type of solid is it?

18. Visualization Th e semicircle is revolved completely about the x-axis.

a. Describe the solid of revolution that is formed.

b. Find its surface area and volume in terms of p.

19. Open-Ended Draw two cones of diff erent heights but the same volume. Label the dimensions of each solid.

20. Error Analysis Your friend tells you that the ratio of the surface area to the volume of a solid is 1 i1. He concludes that the surface area and the volume of this solid are equal. Explain and correct your friend’s error.

2 mm2 mm

2 mm

7 mm

12 cm

8 cm

8 in.

8 in. 10 cm

2 O

2

x

y

2

Chapter 11 Test (continued) Form G

Surface Area and Volume

S.A. 5 45.5 mm2; V 5 12.1 mm3

S.A. 5 150.8 in.2; V 5 116.1 in.3

S.A. 5 336 cm2; V 5 254.0 cm3

S.A. 5 1256.6 cm2; V 5 4188.8 cm3

2578 in.2

17 cm2

If all the circles are O, the shape is a cylinder; if none are O, the shape is a cone of frustum.

a sphere of radius 3 centered at the origin


Answers may vary. Sample: The ratio of the surface area to volume equals 1 unit2 i 1 unit3. So, volume 5 surface area 3 unit. That is, the surface area and volume cannot be equal because area is two-dimensional and volume is three-dimensional.

S.A. 5 36π; V 5 36π


75

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Chapter 11 Quiz 1 Form K


Do you know HOW?

1. Draw a net for the fi gure at the right.

Use the polyhedron at the right for Exercises 3 and 4.

2. Verify Euler’s formula for the fi gure.

3. Draw a net for the polyhedron. Verify F 1 V 5 E 1 1 for the net.

4. Describe the cross section formed in the diagram.

Find the surface area of each fi gure to the nearest whole number.

5. 6.

Do you UNDERSTAND?

7. Th e net of a polyhedron contains six triangles and a hexagon. a. What shape could the polyhedron be?

b. If each triangle has an area of 3 m2 and the hexagon has an area of 6 m2, what is the surface area of the polyhedron?

8. Suppose you build a polyhedron from two octagons and eight rectangles. Without using Euler’s Formula, determine the number of edges the solid has. Explain.

7 cm

7 cm

5 m

12 m

10 1 16 5 24 1 2

10 1 30 5 39 1 1

rectangle

147 cm2 283 m2

24 edges; there are eight edges on each of the 2 octagonal faces, and a total of eight edges connecting the rectangles: 8 1 8 1 8 5 24

hexagonal pyramid

24 m2


76

Name Class Date

Do you know HOW?

Find the volume of the each fi gure to the nearest tenth.

1. 2.

3.

Find the surface area and the volume of each sphere. Give each answer in terms of π.

4. 5.

6. Th e volumes of two similar prisms are 972 cm3 and 36 cm3. Th e surface area of the larger prism is 594 cm2.

a. What is the scale factor for the similar fi gures? b. What is the surface area of the smaller prism?

Do you UNDERSTAND?

7. Vocabulary A solid object has dimensions given in meters. What units might be used to describe the surface area of the object? What units might be used to describe the volume of the object?

8. Error Analysis Two cones have the same volume, but diff erent heights. Th e taller cone is nine times taller than the shorter cone. Your friend concludes that the radius of the shorter cone must be nine times longer than the radius of the taller cone. What is his error?

8 m

7 m

6 m

5 in.

7 in.

9 in.

3 in.

8 in.

5 mm 12 cm

Chapter 11 Quiz 2 Form KLessons 11-4 through 11-7

336 m3 549.8 in.3

72 in.3

S.A. 5 100π mm2 V 5 166 23 π mm3 S.A. 5 144π cm2 V 5 288π cm3

3 i166 cm2

square meters; cubic meters

Answers may vary. Sample: The friend has not considered that the area of the base is B 5 πr2. So, the radius of the shorter cone is only three times as long as the radius of the taller cone.


77

Name Class Date

Chapter 11 Test Form K

Do you know HOW?

1. Draw a net for the fi gure at the right.

2. What is the number of edges in a prism with pentagonal bases?

3. What is the number of vertices in a pyramid with a hexagonal base?

4. Describe the cross section formed in the diagram.

Find the surface area of each fi gure to the nearest whole number.

5. 6.

Find the volume of the each fi gure to the nearest tenth.

7. 8.

9.

10 mm

11 mm 15 in.

40 in.

15 in.

40 in.

5 m

12 m

5 m

4 m

15

7

circle

314.2 m3

8000 in.3

251.3 mm2 3600 in.2

314.2 m3


78

Name Class Date

10. A container company makes small and large cylindrical shampoo containers. Th e area of the base of each cylinder is 10 cm2. Th e volume of the small shampoo container is 60 cm3. Th e large shampoo container is 3 cm taller. What is the volume of the large shampoo container?

Find the surface area and the volume of each sphere. Give each answer in terms of π.

11. 12.

13. Th e circumference of a standard bowling ball is about 27 in. To the nearest unit, how many square inches is the surface area of 100 bowling balls?

For Exercises 12 and 13, write whether the two fi gures are similar. If so, give the scale factor of the fi rst fi gure to the second fi gure.

14. 15.

Do you UNDERSTAND?

16. Error Analysis Your friend draws a net of a prism. What is your friend’s error?

17. Compare and Contrast How are the formula for the surface area of a prism and the surface area of a cylinder alike? How are they diff erent?

18. Reasoning A cylinder has a height of 5 mm and a volume of 80 mm3. What is the radius of the base to the nearest tenth?

11 in. 5 cm

9 ft9 ft

12 ft12 ft

12 ft9 ft

2 m 9 m

6 m

6 m4 m

3 m

Chapter 11 Test (continued) Form K

S.A. 5 484π in.2 V 5 1774 23 π in.3

90 cm3

S.A. 5 25π cm2 V 5 20 56 π cm3

23,205 in.2

yes; 3 i4no

Answers may vary. Sample: They are both of the form S.A. 5 L.A. 1 2B. Both lateral areas involve multiplying the height by the perimeter or circumference. However, the formulas for the areas of the bases are different.

The net is missing one of the bases.

2.3 mm


79

Name Class Date

Performance Tasks

Chapter 11

Task 1Sketch the solids described below. Th en complete steps (c) through (e). a. Sketch a triangular prism. b. Sketch a cylinder of the same height. c. Label both fi gures, giving only the dimensions necessary to calculate the

surface area and the volume. d. Find the surface area and volume of the prism to the nearest tenth. e. Find the surface area and volume of the cylinder to the nearest tenth.

Task 2Compare and contrast pyramids and cones by working through steps (a) through (e). a. Sketch a pyramid and cone. Use variables to label the dimensions. b. Give formulas for the surface area and volume of the pyramid. c. Give formulas for the surface area and volume of the cone. d. Explain the similarities between pyramids and cones. Use your sketches

and formulas. e. Explain the diff erences between pyramids and cones. Use your sketches

and formulas.

Answers may vary. Check students’ work. Sample: a–c. d. prism: S.A. 5 70.6 cm2; V 5 32 cm3

e. cylinder: S.A. 5 31.4 cm2; V 5 12.6 cm3

[4] Student provides accurate sketches and calculations.[3] Student provides sketches and calculations that may contain minor errors.[2] Student provides sketches and calculations that contain some errors.[1] Student provides sketches and calculations that contain signifi cant errors.[0] Student makes little or no progress.

4 cm4 cm

4 cm2 cm

4 cm

Answers may vary. Check students’ work. Sample: a.

b. S.A. 5 B 1 L.A. 5 x2 1 4(12xs) 5 x2 1 2xs; V 5 1

3 Bh 5 13 x2h

c. S.A. 5 B 1 L.A. 5 πr2 1 πrs; V 5 13 Bh 5 1

3πr2h d. Both have one base, a vertex, and volume of

V 5 13Bh

e. Pyramids have polygonal bases and their sides are distinct faces, whereas cones have circular bases and therefore a continuous smooth lateral surface without faces.

[4] Student provides accurate sketches and calculations.[3] Student provides sketches and calculations that may contain minor errors.[2] Student provides sketches and calculations that contain signifi cant errors.[1] Student provides sketches and calculations that contain some errors.[0] Student makes little or no progress.

hr

ss

x

h


80

Name Class Date

Task 3 Sketch and analyze a sphere by working through steps (a) through (e). a. Sketch a sphere and label the diameter d. b. Give formulas for the surface area and volume of the sphere in terms of its

diameter, d. c. Shade a hemisphere on the sphere. d. Give a formula for the surface area of a hemisphere. e. Give a formula for the volume of a hemisphere.

Task 4Explore similar solids by working through steps (a) through (d). a. Are all cubes similar? If so, explain why. If not, give an example of two

cubes that are not similar. b. Are all cones similar? If so, explain why. If not, give an example of two

cones that are not similar. c. Are all spheres similar? If so, explain why. If not, give an example of two

spheres that are not similar. d. Describe a family of pyramids that are similar. (Hint: Use right triangles that

are similar.)

Performance Tasks (continued)

Chapter 11

a. c.

[4] Student provides accurate sketches and formulas.[3] Student provides sketches and formulas that may contain minor errors.[2] Student provides sketches and formulas that contain some errors.[1] Student provides sketches and formulas that contain signifi cant errors.[0] Student makes little or no progress toward the correct sketches or formulas.

d

See below.

S.A. 5 4π(d2)2 5 πd2; V 5 43 π(d2)3 5 1

6 πd3

S.A. 5 12πd2

V 5 112πd3

See below for sample.

Answers may vary. Sample: a. Yes; a scale factor can be found between any two cubes. The scale factor

between two cubes with sides of lengths x and y is x iy. b. no; example: a cone with radius 3 cm and height 5 cm and a cone with radius

6 cm and height 7 cm c. Yes; a scale factor can be found between any two spheres. The scale factor

between two spheres with radii of lengths r and s is r i s. d. pyramids with square bases whose heights are in the same ratio as the side

lengths of the squares[4] Student provides accurate answers and reasons.[3] Student provides answers and reasons that may contain minor errors.[2] Student provides answers and reasons that contain some errors.[1] Student provides answers and reasons that contain signifi cant errors.[0] Student makes little or no progress.


81

Name Class Date

Cumulative Review Chapters 1–11

Multiple Choice

1. What is the value of x in the fi gure?

5 6

5"3 6"3

2. Th e slope of a line is 57. What is the slope of any line that is perpendicular to that line?

2 75 2

57 5

7 75

3. What is the lateral area of the cylinder?

42p cm2 51p cm2

48p cm2 63p cm

4. What is the area of an equilateral triangle with sides of length 6?

9 9"3 18 18"3

5. Th e ratio between the areas of two similar polygons is 3 i4. What is the ratio between the lengths of their corresponding sides?

3 i4 9 i16 "3 i2 32 i2

6. Which fi gure shows nABC after a rotation of 1808 clockwise?

60

x10

7 cm

6 cm

A B

C

A

BC A

B C

AB

C A B

C

A

F

A

G

C

G


82

Name Class Date

Short Response

7. Explain how to construct the perpendicular bisector of line segment AB. Use three steps.

8. Th e area of nABC is 15. Express the altitude from angle B in terms of a, b, and c.

9. Use the diagram at the right to complete the following proof.

Given: Z is the midpoint of AX and BY .

Prove: /A > /X

Statements Reasons

1) /AZB > /XZY

2) Z is the midpoint of AX and BY .

3) 9

4) 9

5) 9

1) 9

2) 9

3) 9

4) 9

5) 9

10. Th e volumes of two similar containers are 56 cm3 and 448 cm3. How many times greater is the surface area of the larger container than to the surface area of the smaller container?

11. Find the area of a regular hexagon with sides of length 8 mm. Round to the nearest tenth of a square millimeter.

A

B

C

a

b

c

AY

BX

Z

Cumulative Review (continued)

Chapters 1–11

Step 1) Put the compass point on A and draw a long arc. Be sure that the opening is greater than 12AB. Step 2) With the same compass setting, put the compass point on B and draw another long arc. Label the two points where the two arcs intersect as X and Y. Step 3) Draw XY, which is the perpendicular bisector of AB.

166.3 mm2

30b

Vertical angles are congruent.

Given

Defi nition of a midpoint

SAS Postulate

Corresponding parts of congruent triangles are congruent.

The larger container is four times the surface area of the smaller container.

AZ O XZ and BZ O YZ

kAZB O kXZY

lA O lX


83

T E A C H E R I N S T R U C T I O N S

Chapter 11 Project Teacher Notes: The Place Is Packed

About the ProjectStudents will explore designs of packages and what a manufacturer might consider when designing the packaging of a product. Th en they will design and construct their own package and shipping container.

Introducing the Project• Ask students: Why are the drinks you buy at a fast-food restaurant sold

in cups that are wider at the top than at the bottom? Why are books rectangular and not round or trapezoid-shaped? Why are many toys in packages unassembled?

• Ask students whether the package design of any products they use has changed (toothpaste containers, soft drink containers, juice boxes). Why do they think the packaging has been changed?

Activity 1: MeasuringCheck that the containers students collect can be cut open. As they cut them open, make sure that they are careful to include all sections of the package in their calculations of the total area of the packaging. Students must know how to calculate the area of cylinders and prisms.

Activity 2: Analyzing Th is activity reinforces the fact that diff erent prisms can have the same volume.

Activity 3: InvestigatingYou may want to provide a variety of containers of diff erent shapes and sizes for students to analyze. Alternatively, students could draw examples of containers that have deceptive shapes.

Activity 4: DesigningEncourage students to complete their package designs by adding the product information on the packages.

Finishing the ProjectYou may wish to plan a project day on which students share their completed projects. Encourage students to share their processes as well as their products.

• Have students review their package designs and shipping containers along with their measurements and calculations.

• Ask students to share any insights they found when completing the project, such as how they decided what package and shipping container designs to use.


84

Name Class Date

Chapter 11 Project: The Place Is Packed

Beginning the Chapter ProjectManufacturers consider dozens of factors before determining which shape will best suit the consumer and boost the company’s profi ts.

In the chapter project, you will explore package design and uncover reasons for the shapes that manufacturers have chosen. You will design and construct your own package and shipping container. You will see how spatial sense and business go hand in hand to determine the shapes of things you use every day.

ActivitiesActivity 1: MeasuringCollect some empty cardboard containers shaped like prisms and cylinders.

• Measure each container, and calculate its surface area.

• Flatten each container by carefully separating the places where it has been glued together. Find the total area of the packaging material used.

• For each container, fi nd the percent by which the area of the packaging material exceeds the surface area of the container.

1. How does an unfolded prism-shaped package diff er from a net for a prism? 2. What did you fi nd out about the amount of extra material needed for

prism-shaped containers? For cylindrical containers? 3. Why would a manufacturer be concerned about the surface area of a

package? About the amount of material used to make the package?

Activity 2: AnalyzingCopy and complete the table below for four diff erent rectangular prisms, each of which has a volume of 216 cm3.

Length (cm)

Width (cm)

Depth (cm)

Volume (V) (cm3)

Surface Area (S.A.) (cm2)

Ratio V : S.A.

6 6 216

2 3 216

3 24 216

6 9 216

1. Which of the prisms uses the container material most effi ciently? Least effi ciently? Explain.

2. Why would a manufacturer be concerned about the ratio of volume to surface area?

3. Why are cereal boxes not shaped to give the greatest ratio of volume to surface area?



6

3

4

36

216

372

306

228

1 i 1

18 i 31

12 i 17

18 i 19

6 by 6 by 6; 2 by 3 by 36; the lower the ratio, the more material is being used to enclose the same volume.

to minimize the cost of packaging

Sample: A cereal box is used for promoting the product, so it needs a large surface area on the front.


85

Name Class Date

Chapter 11 Project: The Place Is Packed (continued)

Activity 3: InvestigatingDo some container shapes seem to contain more product for the same money? Go to a supermarket and identify a variety of container shapes. Do some shapes make you think the containers hold more than they actuallly do? What factors does a manufacturer consider in deciding the shape of a container? Write a report about your fi ndings.

Activity 4: Designing Design and construct your own package for a product of your choice. Draw a net for your package, and specify the dimensions, surface area, amount and type of packaging material used, and volume of the package. Justify your design with mathematical and economic arguments.

Finishing the ProjectTo ship your product, the individual packages must be packed into larger containers. Design and build a container that will effi ciently pack the containers you designed in Activity 4: Designing. Draw a net for your shipping container, including dimensions, surface area, the amount and type of material used, volume of the container, and the number of individual packages that will be packed in the container. Justify your design with mathematical and economic arguments.

Refl ect and ReviseAsk a classmate to review your project with you. Together, check that your package design and shipping container are complete, your diagrams and explanations clear, and your information accurate. Have you used geometric terms correctly? Have you considered other possible designs? Revise your work as needed.

Extending the ProjectFind pictures of packaged products in newspapers or magazine advertisements. Identify the shape of each package. Give possible reasons that the manufacturer chose each package design.




86

Name Class Date

Chapter 11 Project Manager: The Place Is Packed

Getting Started Read about the project. As you work on it, you will need an assortment of empty containers or packages for various products. Keep all of your written work for the project in a folder, along with this Project Manager.

Checklist Suggestions

☐ Activity 1: percent overlap

☐ For example, if the surface area of a box is 75 cm2 and it contains 90 cm2 of cardboard, the amount of cardboard exceeds the surface area by 15

75 of 20%.

☐ Activity 2: effi cient containers

☐ To simplify calculations, choose whole-number values for the dimensions.

☐ Activity 3: deceptive shapes

☐ Include some cone-shaped or tapered containers (syrup, oil, and dish detergent, for example).

☐ Activity 4: package design

☐ Th ink of a product that you fi nd interesting, use a lot, or wish someone would make.

☐ container design ☐ Consider also that many of the containers you design would be packaged for shipping.

Scoring Rubric4 Information is accurate and complete. Diagrams and explanations are clear.

Use of geometric language is appropriate and correct. Design gives exact dimensions, surface area, volume, type, and amount of packaging material. Designs are defended with valid reasoning. Model is well contructed.

3 Information may contain a few minor errors. Diagrams and explanations are understandable. Most geometric language is used appropriately and correctly. Designs are mostly complete, and there is justifi cation for the designs.

2 Much of the information is incorrect. Diagrams and explanations are hard to follow or misleading. Geometric terms are lacking, used sparsely, or misused. Signifi cant portions of the project are unclear or inaccurate.

1 Major elements of the project are incomplete or missing.

0 Project was not handed in, or work does not follow instruction.

Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.

Teacher’s Evaluation of the Project

Documents

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