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CHAPTER 10 THREE-DIMENSIONAL GEOMETRY THEME: History H istory is a record of significant facts, events, and people. The study of history includes architecture, politics, music, population, language, religion, and mathematics. Specifically, the history of mathematics stretches to every part of the world and spans over thousands of years. Three-dimensional geometry has been studied since ancient times. Consequently, people from every culture have required three-dimensional geometry to design shelters, clothing, and storage containers. • Urban planners (page 431) supervise the construction of buildings, roads, bridges, and other structures within a city. Because it is important to maintain historically significant elements, while providing for the needs of a city, history plays its part in a city’s development. • Exhibit designers (page 451) create images and displays that show a product, historical event, or structure that is pleasing to one’s eye. mathmatters2.com/chapter_theme 418 Mathematicians throughout History Mathematician Life span Accomplishment Archimedes c.298 –212 B.C. Greatest mathematician of ancient times. Made contributions in geometry. Easley, Annie 1933– Developed computer code used to identify energy conversion systems for NASA. Euclid c.325 –265 B.C. Wrote The Elements, 13 books covering geometry and number theory. Gauss, Carl Friedrich 1777–1855 Published treatise on Number Theory. Discovered the asteroid Ceres. Hypatia c.370–415 She is considered the first woman of mathematicians. Khayyam, Omar c.1048–1131 Persian mathematician, astronomer and poet. Revised the Arabic calendar. Solved cubic equations through geometry. Lovelace, Ada Byron 1815–1852 Suggested that Babbage's first "computer" calculate and play music. Pascal, Blaise 1623–1662 Invented and sold the first adding machine in 1645. Developed probability theory. Pythagoras of Samos c.560 –c.480 B.C. His theorem showed the existence of irrational numbers. (c. stands for circa, meaning about or approximately) Data Activity: Mathematicians throughout History Use the table for Questions 1–4. 1. How long did Pythagoras live? 2. Who is considered the greatest mathematician of ancient times? 3. How many of these mathematicians did work with number theory? Who were they? 4. List the names of mathematicians in the table with whom you are familiar. Discuss with a partner one or two other accomplishments credited to each person on your list. Gauss CHAPTER INVESTIGATION Many people have contributed to the modern understanding of geometry. By studying the work of early mathematicians, you can develop a better understanding of mathematical concepts. Working Together Design a timeline of significant mathematicians who made contributions to geometry. Use the timeline to understand how mathematics influenced life throughout history. Use the Chapter Investigation icons to guide you to a complete timeline. Pascal Chapter 10 Three-Dimensional Geometry 419 10 Are You Ready? CHAPTER Refresh Your Math Skills for Chapter 10 The skills on these two pages are ones you have already learned. Use the examples to refresh your memory and complete the exercises. For additional practice on these and more prerequisite skills, see pages 576–584. PERIMETER FORMULAS In this chapter you will work with three-dimensional solids. Formulas that apply to two-dimensional shapes can be a foundation for understanding solid geometry. Perimeter of a square Perimeter of a rectangle Perimeter of any polygon P 4s, where s side P 2l 2w, where l length P sum of the lengths of length and w width the sides Examples Find the perimeter of each figure. 7 ft 8 ft P 2(8) 2(5) P 4 7 8 6 9 P 16 10 4 ft P 34 ft 5 cm P 26 cm 6 ft 8 cm 9 ft Find the perimeter of each figure. 1. 2. 5 in. 3. 8 in. 7 cm 7 in. 12 m 15 m 5 in. 12 cm 18 m 12 in. 9 in. 4. 5. 6.5 m 6.5 m 6. 18 yd 22 cm 11 yd 9 yd 6.5 m 6.5 m 24 yd 22 cm 6.5 m AREA FORMULAS Area of a rectangle or square Area of a parallelogram Area of a triangle 1 A l w, where l length A b h, where b length A bh, where b length 2 and w width of the base and h height of the base and h height 420 Chapter 10 Three-Dimensional Geometry Examples Find the area of each figure. 1 A 15 7 A (20)(15) 7 mm 2 15 in. 1 A 105 mm2 A (300) 15 mm 2 A 150 in.2 20 in. Find the area of each figure. 7. 8. 9. 23 cm 4 in. 13 ft 16 cm 5 in. 13 ft 10. 11. 12. 12 in. 15 cm 32 m 17 in. 45 m 9 cm VOLUME FORMULAS Volume of a rectangular prism Volume of a triangular prism V B h, where B area of the V B h, where B area of the base and h height base and h height Examples Find the area of the base and substitute that measure to find the volume. B 3 4 1 B (5)(8) 2 8 cm B 12 1 B (40) V 12 8 2 9 ft 3 cm B 20 3 V 96 cm 4 cm 5 ft V 20 9 180 ft3 8 ft Find the volume of each figure. 13. 14. 15. 9 yd 10 m 8m 12 mm 13 m 3 yd mm 8 yd 22 9 mm Chapter 10 Are You Ready? 421 10-1 Visualize and Represent Solids Goals I I Identify properties of three-dimensional figures. Visualize three-dimensional geometric figures. Applications Packaging, History, Sports, Machinery, Recreation Use a pencil, paper, tape and scissors. 1. Cut a small rectangle, a small right triangle and a small semicircle out of the paper. 2. Tape the rectangle near the top of the pencil. Spin the pencil on the tip to visualize what solid is created by spinning the rectangle. Describe the solid. 3. Repeat Question 2 for the right triangle. 4. Repeat Question 2 for the semicircle. BUILD UNDERSTANDING Some three-dimensional figures have both flat and curved Axis Axis surfaces. A cylinder has a curved surface and two congruent circular Right Oblique bases that lie in parallel planes. The axis is a segment that cylinder cylinder joins the centers of the bases. If the axis forms a right angle with the bases, it is a right cylinder. If not, it is an oblique cylinder. Axis Axis A cone has a curved surface and one circular base. The axis is a segment that joins the vertex to the center of the base. If the axis Oblique Right forms a right angle with the base, it is a right cone. Otherwise, it is cone cone an oblique cone. Center A sphere is the set of all points that are a given distance from a given point, called the center of the sphere. A polyhedron (plural: polyhedra) is a closed, three- dimensional figure made up of polygonal surfaces. Sphere The polygonal surfaces are faces. Two faces meet, or Face Base intersect, at an edge. A point at which three or more Edge edges intersect is a vertex. Vertex A prism is a polyhedron with two identical parallel faces called bases. The other faces are parallelograms. A prism is named Base Pentagonal prism according to the shape of its base. Triangular Vertex A pyramid is a polyhedron with only one base. pyramid The other faces are triangles that meet at a vertex. Face A pyramid is named by the shape of its base. Base 422 Chapter 10 Three-Dimensional Geometry In prisms and pyramids, the faces that are not bases are lateral faces. The edges of these faces are lateral edges. Lateral faces can be Check either parallel or intersecting. The lateral edges can be intersecting Understanding or parallel. Skew edges are noncoplanar. Why is a cube also a rectangular prism? Right Oblique Right Oblique rectangular rectangular square square prism prism pyramid pyramid Example 1 PACKAGING Identify each feature of the shoe box. B C a. the shape of the shoe box b. a pair of bases A c. a pair of parallel edges d. a pair of intersecting edges D H Style A5 G Brown e. a pair of skew edges f. a pair of intersecting faces Size 9 E F Solution a. The shoe box is a right rectangular prism. b. Two bases are ADFE and BCHG. c. DC and FH are parallel edges. d. GH and FH are intersecting edges. e. AD and CH are skew edges. f. ABCD and CDFH are intersecting faces. Example 2 Draw a right triangular prism. Solution Step 1 Draw two congruent triangles. Step 2 Draw segments that connect the corresponding vertices of the triangles. Use dashed segments to show the edges that cannot be seen. TRY THESE EXERCISES Identify each figure and name its base(s). 1. A B 2. S F E D C T U H G V mathmatters2.com/extra_examples Lesson 10-1 Visualize and Represent Solids 423 If possible, identify a pair of parallel faces, parallel edges, intersecting faces, and intersecting edges for each figure. 3. the figure in Exercise 1 4. the figure in Exercise 2 Draw each figure. 5. right circular cone 6. oblique rectangular prism 7. right hexagonal prism 8. right square pyramid PRACTICE EXERCISES • For Extra Practice, see page 616. Identify each figure and name its base(s). 9. E 10. A B 11. G L H K D A C F G I F E J F B E D H H G C D Identify a pair of parallel faces, parallel edges, intersecting faces, and intersecting edges for each figure. 12. the figure in Exercise 9 13. the figure in Exercise 10 14. the figure in Exercise 11 Draw each figure. 15. right rectangular prism 16. oblique square pyramid 17. oblique cone 18. right cylinder 19. sphere 20. right triangular prism 21. WRITING MATH Describe ways in which prisms and pyramids are similar. Describe differences between prisms and pyramids. RECREATION If possible, identify the geometric figure represented and give the number of faces, vertices and edges. 22. tennis ball can 23. six-sided number cube 24. basketball 25. ice cream cone 26. rectangular gym 27. one-person tent 28. Which of the figures in Exercises 22–27 are polyhedra? Explain. MACHINERY A cutting drill uses different bits to cut designs into furniture. Describe the geometric figure cut when using each of the following bits. 29. 30. 31. 32. YOU MAKE THE CALL Niki says that a right square pyramid is a polyhedron but an oblique square pyramid is not a polyhedron. Daniella says both figures are polyhedra since their faces are polygonal. Who is correct and why? 424 Chapter 10 Three-Dimensional Geometry Decide whether each statement is true or false. Explain. 33. A cube is a polyhedron. 34. A polyhedron can have exactly three faces. 35. A polyhedron can have a circular face. 36. A polyhedron can have exactly four faces. State whether each solid is a polyhedron. Explain. 37. 38. 39. 40. HISTORY The pyramids of Giza in Egypt are one of the seven wonders of the ancient world. Use the photo on page 433 to sketch the pyramid on paper. Then label the vertices. EXTENDED PRACTICE EXERCISES SPREADSHEET Create a spreadsheet for the following figures. List figures in Column A, the number of faces (F ) in Column B, the number of vertices (V ) in Column C, and the number of edges (E ) in Column D. In Column E, write a formula to calculate the quantity F V E . 41. rectangular prism 42. hexagonal prism 43. pentagonal pyramid 44. square pyramid 45. Compare your results for Exercises 41–44. Make a generalization about the relationship between the numbers of faces, edges and vertices in a polyhedron. 46. CHAPTER INVESTIGATION Research ten mathematicians. Record the name, accomplishment, dates of birth and death, date of accomplishment and country of birth for each mathematician. MIXED REVIEW EXERCISES Find the slope of the line containing the given points. Identify any vertical or horizontal lines. (Lesson 6-2) 47. (4, 2), (6, 1) 48. (9, 3), ( 2, 4) 49. ( 1, 3), (2, 8) 50. (5, 4), (7, 7) 51. ( 3, 5), ( 3, 2) 52. (4, 5), ( 2, 3) 53. ( 2, 6), (4, 6) 54. ( 1, 1), ( 7, 7) In the figure, RS MN and m 4 80°. Find each measure. (Lesson 5-3) 55. m 1 56. m 2 R 7 8 S 57. m 3 58. m 5 5 6 59. m 6 60. m 7 3 4 M 1 2 N mathmatters2.com/self_check_quiz Lesson 10-1 Visualize and Represent Solids 425 10-2 Nets and Surface Area Goals I Draw nets for three-dimensional figures. I Use nets to find the surface area of polyhedra. Applications Retail, History, Machinery, Architecture Use grid paper and scissors. 1. Copy the figure shown on grid paper, cut it out and fold along the dotted lines. 2. What type of polyhedron is formed? 3. Draw a different arrangement of the six squares so when you cut and fold the figure it makes the same solid as the one created in Question 2. 4. Test the figure drawn in Question 3. 5. Discuss with your classmates those figures that did create the same solid and those that did not. BUILD UNDERSTANDING A net is a two-dimensional pattern that can be folded to form a three-dimensional figure. A three-dimensional figure can have more than one net. Example 1 What three-dimensional figure is represented by each net? a. b. c. Solution Visualize each net being folded to form a three-dimensional figure. a. This figure will have four rectangular sides and two parallel square bases. It is a rectangular prism. b. This figure will have three triangular sides and a triangular base. It is a triangular pyramid. c. This figure will have three rectangular sides and two parallel triangular bases. It is a triangular prism. 426 Chapter 10 Three-Dimensional Geometry Example 2 Problem Solving Draw a net for each three-dimensional figure. Tip a. right square pyramid If you are unable to figure out the three- b. right cylinder dimensional figure a net represents, copy the net, cut it out and fold it to Solution find the answer. a. b. Nets can be used to calculate the surface area of certain three-dimensional figures. The surface area of a figure is the sum of the areas of all its bases and faces. Example 3 RETAIL Find the minimum amount of gift wrapping paper needed to cover a box with dimensions 1 ft by 2 ft by 3 in. Solution 3 in. Draw the net for the rectangular prism, and calculate the areas of the six sides. Use the formula A l w. The top and bottom of the box are rectangles with dimensions 1 ft by 2 ft. 2 ft l w l w Find the combined surface area. 2 1 2 1 4 ft2 The left and right sides of the box are rectangles with dimensions 2 ft by 3 in. 1 ft 3 in. 0.25 ft Convert 3 in. to feet. 1 ft 12 in. l w l w Find the combined surface area. 2 0.25 2 0.25 1 ft2 Check The front and back of the box are rectangles with dimensions 1 ft Understanding by 3 in. Use 0.25 ft for 3 in. In Example 3, why are l w l w you asked to find the 2 1 0.25 1 0.25 0.5 ft minimum amount of wrapping paper needed The total surface area is the sum of the areas of the sides. for the box? 4 + 1 + 0.5 5.5 ft2 So the box will require at least 5.5 ft2 of gift wrapping paper. mathmatters2.com/extra_examples Lesson 10-2 Nets and Surface Area 427 TRY THESE EXERCISES Identify the three-dimensional figure for each net. 1. 2. 3. 4. Draw a net for each three-dimensional figure. 5. cube 6. pentagonal prism 7. rectangular prism Draw a net for each figure. Then find the surface area. Round each answer to the nearest tenth. Use 3.14 for π. 8. 9. 10. 4.2 ft 23.5 cm 0.5 m 4m 2m 10 ft 3 ft 11.1 cm 3 ft PRACTICE EXERCISES PRACTICE EXERCISES • For Extra Practice, see page 616. Identify the three-dimensional figure for each net. 11. 12. 13. Draw a net for each three-dimensional figure. 14. octagonal prism 15. triangular prism 16. right cone 17. HISTORY Design and draw a Native American tepee. Then draw a net for the tepee. 18. Draw a net of a cube that has a side length of 6 m. Then find the surface area of the cube. 19. ARCHITECTURE What is the total surface area of the glass 5 ft (including the roof) used to build the greenhouse? Draw a net 3 ft of the walls and the roof to solve the problem. 6 ft 10 ft 428 Chapter 10 Three-Dimensional Geometry 8 ft Find the area of each net. Use 3.14 for π. Round to the nearest tenth if necessary. 20. 21. 5 ft 22. The height of each triangle 5 3 is . 3 4 cm 8 ft 4 cm 15.7 ft 3 1 ft 3 23. WRITING MATH How can nets help you find the surface area of three- dimensional figures? What kinds of mistakes do you make when calculating surface area? EXTENDED PRACTICE EXERCISES Use nets to find the surface area of each prism. Use 3.14 for π. 24. 12 in. 25. 7m 26. 2.5 ft 9 in. 10 ft 7 in. 6 in. 10 in. 3m 8 in. 27. MACHINERY The wheel of a steamroller is a cylinder. If the wheel’s diameter is 5 ft and its width is 7.2 ft, approximately how much area is covered by one complete wheel revolution? Use a net to visualize and solve the problem. Round the answer to the nearest whole number. 28. CRITICAL THINKING There are 11 different nets for a cube. Draw as many of them as you can. 29. Refer to exercise 28. To construct 20 cubes out of construction paper, which of the 11 nets would you use? Explain. MIXED REVIEW EXERCISES Simplify each numerical expression. (Lesson 2-2) 30. 4 3 6 4 31. 8 6 42 3 32. 2 (5 2)2 4 33. 15 3 (7 3) 2 34. 4 8(3 4) 2 35. (6 1)2 15 3 4 Evaluate each expression when c 11. (Lesson 2-2) 36. 18 c 37. 3(c 8) 38. (c 8)2 4 39. 9 c(3c 2) Simplify. (Lesson 9-2) 40. (4r) (9p) 41. (6r 2) (4r 3g 2) 42. ( 3a 2) (5ab 2)2 43. (5c 3d) (2cd 4) 44. ( 4k 4f 2) ( 6kf 6) 45. (3s 3t 2) ( 6st 3q) mathmatters2.com/self_check_quiz Lesson 10-2 Nets and Surface Area 429 Review and Practice Your Skills PRACTICE LESSON 10-1 Choose a word from the list to complete each statement. ?__ 1. A ___ has one square base and triangular faces. a. cone ?__ 2. A ___ has a curved surface and one circular base. b. cylinder ?__ 3. A ___ has two congruent circular bases that lie in parallel planes between a curved surface. c. sphere 4. A ___ is a figure consisting of the set of all points in space ?__ d. square pyramid that are a given distance from a given point. e. triangular prism ?__ 5. A ___ has a triangular base and triangular faces. f. triangular pyramid ?__ 6. A ___ has two triangular bases parallel to each other with the faces that are parallelograms. Identify each figure. State the number of faces, vertices and edges of each. 7. 8. 9. PRACTICE LESSON 10-2 Identify the three-dimensional figure for each net. 10. 11. 12. Draw a net for each three-dimensional figure. 13. 14. 15. 4 cm 12 cm 6.5 cm 3 cm 4 in. 5 cm 6 in. 4 cm 3 in. Find the surface area of each figure. 16. the figure in Exercise 13 17. the figure in Exercise 14 18. the figure in Exercise 15 430 Chapter 10 Three-Dimensional Geometry PRACTICE LESSON 10-1–LESSON 10-2 Draw a net for each three-dimensional figure. (Lesson 10-2) 19. 20. 21. 22. Draw a triangular pyramid. 23. How many vertices, faces and edges does a triangular pyramid have? Match each pair of edges of the cube with an item in the box. (Lesson 10-1) A B a. parallel edges 24. AD and CG D C b. skew edges 25. BC and EH E F c. perpendicular 26. CG and GH edges H G Career – Urban Planner Workplace Knowhow A n urban planner designs and supervises the construction of roads, buildings, tunnels, bridges, and water supply systems. The city of Berlin, Germany, changed rapidly following the removal of the Berlin Wall. New construction was needed to reconstruct the city and bring its buildings to modern standards. An urban planner in Berlin must decide what types of buildings are most appropriate for the needs of the city while maintaining the historical look and structure. Suppose an urban planner is beginning the design process for a new government building in Berlin. 1. The shape of this new building is a right hexagonal prism. Make a sketch of the new structure, drawing the regular hexagons at the base and the top of the building. 2. Draw a net (to scale) of the building. The sides of the regular hexagonal bases should have a length of 30 m, and the height of the building should be 60 m. 3. What is the total surface area of the walls? 4. How many marble tiles will be required to cover the outside walls if the tiles are 60 cm wide and 10 cm tall? mathmatters2.com/mathworks Chapter 10 Review and Practice Your Skills 431 10-3 Surface Area of Three- Dimensional Figures Goals Applications I Find the surface area of three-dimensional figures. Geography, Food service, Packaging, History Use a box such as a cereal box or a shoe box for the activity. 1. Carefully tear the box apart at the edges. Do not keep the tabs that were used to glue the faces of the box together. 2. How many polygons do you now have? Describe them. 3. Use a ruler to measure. Then calculate the area of each of the polygons. 4. Add the area of all the polygons. What does the total area represent? BUILD UNDERSTANDING In the previous lesson, you learned that the surface area of a polyhedron is the sum of the areas of all its bases and faces. You saw how the net of a three- dimensional figure can be used to find the total surface area. In this lesson, you will use formulas to find the surface area of three-dimensional figures. Example 1 A 4 in. B Find the surface area of the rectangular prism. C 10 in. 6 in. Solution A rectangular prism has three pairs of congruent faces. The surface area is the sum of the areas of all the faces. Use the formula SA 2(area A) 2(area B) 2(area C), where each area can be found using the formula for the area of a rectangle, A lw. area of A A 10 6 A 6 in. A 60 10 in. area of B A 10 4 A 40 B 4 in. Check 10 in. Understanding area of C A 6 4 C 4 in. How is using the surface A 24 6 in. area formula similar to using nets to find the SA 2 (area A) 2 (area B) 2 (area C) surface area of a three- SA 2 60 2 40 2 24 dimensional figure? SA 248 The surface area is 248 in.2. Chapter 10 Three-Dimensional Geometry 432 A square pyramid with four congruent triangular faces is called a regular square pyramid. If you know the dimensions of just one triangular face, you can calculate the surface area of the pyramid. Example 2 GEOGRAPHY The Great Pyramid at Giza in Egypt has a square base approximately 756 ft long. The height of each triangular face, called the slant height, is approximately 612 ft. What is the approximate surface area of the faces of the Great Pyramid? Solution The Great Pyramid is a regular square pyramid, so its faces are four congruent triangles. Calculate the surface area of one of the faces and multiply by 4. 1 A bh Use the formula for the area of a triangle. 2 1 A 756 612 231,336 2 SA 4 231,336 925,344 The surface area of the faces of the Great Pyramid at Giza is approximately 925,344 ft2. A cylinder is a figure with two congruent circular bases and a curved surface. To find the surface area, add the areas of the bases to the area of the curved surface. Surface Area SA 2 rh 2 r2 of a Cylinder where r is the radius of a base and h is the height of the cylinder. Example 3 10 cm Find the surface area of the cylinder. 25 cm Solution SA 2 rh 2 r2 SA 2 3.14 10 25 2 3.14 102 3.14 SA 1570 628 2198 The surface area of the cylinder is approximately 2198 cm2. A cone is a three-dimensional figure with a curved surface and one circular base. The height of a cone is the length of a perpendicular slant height height segment from its vertex to its base. The slant height (s) of a cone is the (s) length of a segment from its vertex to its base along the side of the cone. Surface SA rs r2 Area of a where r is the radius of the base and s is the slant r Cone height. mathmatters2.com/extra_examples Lesson 10-3 Surface Area of Three-Dimensional Figures 433 Example 4 10 cm Find the surface area of the cone. Solution SA rs r2 30 cm SA 3.14 10 30 3.14 102 3.14 SA 942 314 SA 1256 The surface area of the cone is approximately 1256 cm2. TRY THESE EXERCISES Find the surface area of each figure. Round to the nearest hundredth. 1. 2. 17 m 3. 9 ft 4. 2.8 m 10 m 4.3 m 8 ft 22 ft 17 ft 42 m 6 ft 21 m 8m 5. 6. 24 m 7. 0.4 cm 8. 7.2 mm 10 m 1 cm 25 ft 26 m 22.5 ft 10 m 20 mm 22.5 ft 3.6 cm PRACTICE EXERCISES • For Extra Practice, see page 617. Find the surface area of each figure. Round to the nearest hundredth. 9. 10. 9.2 cm 11. 12. 12 m 1.8 m 5m 0.9 m 1.1 m 26 cm 1.1 m 4.5 m 30 m 13 m 13. 14. 15. 16. 20 ft 10 m 14.5 in. 4m 19 in. 13 m 22 ft 19 in. 24 ft 10 m 6.6 m 17. The base of each triangular face of a regular square pyramid is 9 in., and the height of each triangular face is 11 in. Find the surface area of the pyramid. 434 Chapter 10 Three-Dimensional Geometry 18. PACKAGING Tamara is wrapping a gift. The box is 14 in. by 9 in. by 8 in. What is the minimum amount of wrapping paper she will need to cover the box? 19. WRITING MATH Explain why the word “minimum” is used in Exercise 18. 20. FOOD SERVICE Find the surface area of a can of cranberry sauce that has a height of 4.2 in. and a radius of 1.5 in. 21. SPREADSHEET Use a spreadsheet to calculate the surface area of cylinders. Enter the formula for the surface area of a cylinder into the first cell. Then evaluate the formula for a cylinder that is 12 cm long with a radius of 3 cm, 6 cm, 12 cm, and 24 cm. 22. HISTORY An ancient Mayan pyramid has a square base. Each side of the base measures 230 m, and the slant height is 179 m tall. Find the area of the faces of the Mayan pyramid. EXTENDED PRACTICE EXERCISES Find the surface area of each figure. 23. 4m 24. 6m 25. 10 m 8m 4m 12 m 7m 12 m 12 m 4m 26. CRITICAL THINKING A cone has a height h and a radius r. Find an expression for the slant height. Then write a formula for the surface area of the cone using this expression. 27. A hole is drilled through a solid cube that has edges of 4 cm. The hole is drilled perpendicular to the top face of the cube. The diameter of the hole is 2 cm. To the nearest whole number, what is the total surface area of the resulting solid figure? 28. DATA FILE Refer to the data on U.S. shopping centers on page 570. What are possible dimensions of a rectangular prism whose surface area is equal to the area of Tysons Corner Center? MIXED REVIEW EXERCISES Write the equation of each line using the given information. (Lesson 6-3) 29. m 3, (1, 4) 30. m 2, ( 2, 3) 31. (2, 7), (3, 9) 32. (0, 2), (2, 4) Find each product. (Lesson 9-5) 33. (r 3) (r 4) 34. (z 1) (z 4) 2 35. (b 4) 36. (f 2) (f 5) 37. (v 4) (v 8) 38. (w 1) (w 7) 2 39. (3k 1) (2k 4) 40. (2c 3) mathmatters2.com/self_check_quiz Lesson 10-3 Surface Area of Three-Dimensional Figures 435 10-4 Perspective Drawings Goals I Make one- and two-point perspective drawings. I Locate the vanishing points of perspective drawings. Applications Art, Interior design, History, Architecture Use the three photographs for Questions 1–3. 1. Can you see depth in each of the photographs? 2. Do you think the photographer was located above, below or at eye level with the subject? 3. Does your eye travel to one point or two points in the background of each photograph? BUILD UNDERSTANDING A perspective drawing is a way of drawing objects on a flat surface so that they look the same as they appear in real life. The eye and the camera are both constructed so that objects appear progressively smaller the farther away they are. Parallel lines drawn in perspective appear to come together in the distance. Perspective drawings use vanishing points. A vanishing point is a point that lies on the horizon line. The horizon line is a line in the distance where parallel lines appear to come together. Perspective drawings can be made in either one-point perspective or two-point perspective. Example 1 Draw a cube in one-point perspective. Solution Step 1 Draw a square to show the front surface of the cube. Draw a horizon line j and a vanishing point A on line j. Step 2 Connect the vertices of the square to the vanishing point. A A A j j j Step 1 Step 2 Step 3 Step 4 Step 3 Draw a smaller square whose vertices touch the four line segments. Step 4 Connect the vertices of the two squares. Use a dashed segment to indicate the edges of the cube hidden from view. Remove line j and point A. 436 Chapter 10 Three-Dimensional Geometry A two-point perspective drawing has two vanishing points. If you are looking at the corner of an object in a perspective drawing, then it Math: Who, is probably a two-point perspective drawing. Where, When Ancient Greeks and Romans developed techniques for drawing Example 2 objects in perspective. Renaissance artists in the early 1400s used ARCHITECTURE Draw a building shaped like a rectangular prism perspective drawings to in two-point perspective. create lifelike works. Solution Step 1 Use a straightedge to draw a vertical line VP VP segment to represent the height of the building. Draw a vanishing point (VP) on each side of the segment. Sketch depth lines from the top and bottom of the segment to each point. Step 2 Draw two segments parallel to the original VP VP segment to represent the height. Draw two segments to complete the top of the building. Step 3 Draw two more depth lines from the top corners to the vanishing points. Step 4 Erase the unused portions of the depth lines to complete the two-point perspective drawing. Example 3 Locate the vanishing point of the perspective drawing. Solution To locate the vanishing point, use a straightedge to draw the depth lines from the top edges of the figure. The point of their intersection is the vanishing point of the perspective drawing. TRY THESE EXERCISES Sketch each object in one-point perspective. 1. cube 2. triangular prism 3. rectangular prism 4. cylinder 5. your first name written in capital block-style letters Sketch each object in two-point perspective. 6. cube 7. shoe box 8. table mathmatters2.com/extra_examples Lesson 10-4 Perspective Drawings 437 Trace each perspective drawing to locate the vanishing point(s). 9. 10. 11. PRACTICE EXERCISES • For Extra Practice, see page 617. Sketch each object in one-point perspective. 12. window 13. wall of a room 14. couch Sketch each object in two-point perspective. 15. television 16. street sign 17. school bus Trace each perspective drawing to locate the vanishing point(s). 18. 19. 20. 21. WRITING MATH Describe the location of the vanishing point when a viewer can see three sides of a box drawn in one-point perspective. 22. HISTORY The Sears Tower in Chicago is the tallest building in the U.S. Use a picture of the Sears Tower to make the one-point perspective drawing. 23. INTERIOR DESIGN Make a one-point perspective drawing of the square tile pattern of a floor. A square is cut out of cardboard, and a flashlight shines behind it. Decide if it is possible to obtain the indicated shadow. 24. square 25. segment 26. triangle 27. point 28. trapezoid 29. kite 30. pentagon 31. parallelogram 438 Chapter 10 Three-Dimensional Geometry 32. ARCHITECTURE Describe the perspective used by the architect to create the illusion of a three- dimensional house in her plans. 33. Trace the house on a piece of paper. Locate a vanishing point. 34. GEOMETRY SOFTWARE Use geometry software to make a one- or two-point perspective drawing. Begin by drawing a figure such as a rectangle and locating one or two vanishing points. Sketch the depth lines, and use them to complete the perspective drawing. EXTENDED PRACTICE EXERCISES Determine the number of vanishing points used to draw each cube. 35. 36. 37. 38. RESEARCH Use your school’s library or another resource to find an example of a painting that uses one- and two-point perspectives. Identify the vanishing points used by the artists. 39. CRITICAL THINKING Make a two-point perspective drawing of a rectangular box with a square window cut out of one of the faces. Show that the box has depth or thickness. 40. CHAPTER INVESTIGATION Draw a line segment about 10 in. long across the center of a piece of paper with an arrowhead at each end. Place on the timeline in chronological order the name and accomplishment of each mathematician you researched. Label appropriately. Be sure the distances between the years recorded on your timeline are in proportion. MIXED REVIEW EXERCISES State if it is possible to have a triangle with sides of the given lengths. (Lesson 5-4) 41. 8, 5, 4 42. 10, 6, 4 43. 12, 7, 6 44. 9, 5, 5 45. 16, 14, 12 46. 11, 8, 5 47. 7, 7, 2 48. 12, 6, 5 Factor each polynomial. (Lesson 9-7) 49. 15z 2 18z 50. 35r 2 60r 5 4 3 51. 8y 24y 28 52. 6r s 9r 3s 2 18r 2s 2 53. 28b 3c 84b 2c 5 54. 24wx 3 12w 2x 2 20w 3x 3 55. 24g 4h 40g 2h 2 16g 5 56. 30a 3b 2 45a 2b 3 60a 3b 3 mathmatters2.com/self_check_quiz Lesson 10-4 Perspective Drawings 439 Review and Practice Your Skills PRACTICE LESSON 10-3 Find the surface area of each figure. 1. 8m 2. 1.2 cm 3. 2 mm 6m 5 cm 10 m 12 mm 6m 13 cm 4. 6 cm 5. 6. 4m 3m cm 7m 20 4m 10 m 4m 5m 7. A regular square pyramid has a base area of 100 cm2 and a height of 12 cm. What is the surface area? (Hint: Find the slant height s.) 8. A can of peaches has a height of 5 in. and a diameter of 3 in. How much metal is needed to make the can? PRACTICE LESSON 10-4 Sketch each object in one-point perspective. 9. railroad track 10. house 11. door Sketch each object in two-point perspective. 12. stereo speaker 13. grandfather clock 14. building Trace each perspective drawing to locate the vanishing point(s). 15. 16. 17. 1s tS t 18. Explain the difference between a one-point perspective drawing and a two-point perspective drawing. 440 Chapter 10 Three-Dimensional Geometry PRACTICE LESSON 10-1–LESSON 10-4 Draw a net for each three-dimensional figure. (Lesson 10-2) 19. 20. 21. Refer to the figure. (Lesson 10-1) 22. How many vertices does the figure have? 23. How many faces does the figure have? 24. Name the figure. Mid-Chapter Quiz Identify each figure and name its base(s). (Lesson 10-1) 1. T 2. F 3. I J Q H K S B M P A M N U C R L O E D Identify the three-dimensional figure for each net. (Lesson 10-2) 4. 5. 6. Find the surface area of each figure. (Lesson 10-3) 7. 8. 9. 7 cm 6 in. 7 1 in. 15 m 2 5 cm 12 m 8 in. 12 m Sketch each object in one-point perspective. (Lesson 10-4) 10. pentagonal prism 11. building 12. highway Chapter 10 Review and Practice Your Skills 441 10-5 Isometric Drawings Goals I Visualize and represent objects with isometric drawings. Applications Art, Recreation, History, Architecture Work with a partner. Use several cubes, preferably the kind that attach to each other. 1. Join or stack the cubes together to build a structure. Hold the structure in your hand, and rotate it to obtain different views. 2. Describe the structure to your partner. Which single view is easiest to describe? 3. Make a new structure. Describe it to your partner so that it is clear how to build the same structure. 4. Have your partner build the new structure based on your description. BUILD UNDERSTANDING In the last lesson, you learned how to reproduce three-dimensional figures on paper using perspective drawings. Another way to show a three-dimensional object is with an isometric drawing. An isometric drawing shows an object from a corner view so that three sides of the object can be seen in a single drawing. The figure shows an isometric drawing of a structure. In an isometric drawing, all parallel edges of the structure are shown as parallel line segments in the drawing. Perpendicular line segments do not necessarily appear perpendicular. Example 1 Problem Solving Make an isometric drawing of a cube. Tip The length of all edges in an isometric drawing Solution are either the true length or a scaled length. Step 1 Begin by drawing a vertical line, which will be the front edge Angle measurements are of the cube. Then draw the right and left edges of the cube. not preserved in isometric drawings. Step 2 Draw the vertical sides of the cube parallel to the front edge. Complete the isometric drawing by sketching the rear edges of the cube. 442 Chapter 10 Three-Dimensional Geometry Example 2 Make an isometric drawing of a figure containing three cubes that form an L shape. Solution Step 1 Begin by drawing the left edges of the figure. Then draw the segments for the right edges of the figure. These segments are parallel to the edges of the left side and shown here in blue. Step 2 Complete the isometric drawing by sketching the five remaining segments that outline the cubes. Example 3 Use the isometric drawing to answer the questions. a. How many cubes are used in the drawing? b. How many cube faces are in the figure? c. If each cube face represents 3.5 ft2, what is the total surface area of the figure? Solution a. Even though there are only three visible cubes, the figure contains four cubes. The top cube is resting on a cube beneath it. b. To count the total number of cube faces, proceed in a logical manner so that you do not miss any sides. Cube faces pointing up 3 Cube faces pointing to the front 6 Cube faces pointing to the back 6 Cube faces on the bottom 3 There are a total of 3 6 6 3 18 cube faces on the figure. c. Since there are 18 cube faces and each face represents 3.5 ft2, the total surface area of the figure is 18 3.5 ft2 63 ft2. TRY THESE EXERCISES Create an isometric drawing of each figure. 1. cube 2. rectangular prism 3. triangular prism 4. staircase composed of cubes 5. ARCHITECTURE Using isometric grid paper, design a building using a rectangular prism for the base and a triangular prism for the roof. 6. Use isometric grid paper to draw three different rectangular solids using a total of eight cubes each. mathmatters2.com/extra_examples Lesson 10-5 Isometric Drawings 443 Use the isometric drawing for Exercises 7–9. 7. How many cubes are used in the drawing? How many are hidden? 8. How many cube faces are exposed in the figure? 9. If each face represents 1 yd2, what is the total surface area of the figure? PRACTICE EXERCISES • For Extra Practice, see page 618. Create an isometric drawing of each figure. 10. hexagonal prism 11. figure having 14 faces 12. pentagonal prism 13. figure composed of 11 cubes 14. capital block-style letter M 15. cube on top of a rectangular prism 16. Use isometric grid paper to make a sketch of the letters of your first name. 17. Use isometric grid paper to make a sketch of a rectangular prism with 18 faces. Use the isometric drawing for Exercises 18–20. Assume that no cubes are hidden from view. 18. How many cubes are used in the drawing? 19. How many cube faces are exposed in the figure? 20. If the length of an edge of one of the cubes is 3.4 m, what is the total surface area of the figure? Give the number of cubes used to make each figure and the number of cube faces exposed. Be sure to count the cubes that are hidden from view. 21. 22. 23. 24. 25. 26. 27. RECREATION Make an isometric drawing of a tent. Use a rectangular prism for the base and a triangular prism for the top of the tent. 28. WRITING MATH Explain the similarities and differences between regular square grid paper and isometric dot paper. Which do you prefer to use for drawing? 444 Chapter 10 Three-Dimensional Geometry 29. HISTORY Make an isometric drawing of the ancient Greek Parthenon. 30. YOU MAKE THE CALL Ken says that it is impossible to make an isometric drawing of a rectangular prism containing 6 cubes that has exactly 24 cube faces showing. Brandon says that you can do this by making one long stack of cubes. Who is correct and why? EXTENDED PRACTICE EXERCISES How many cubes are used to make each isometric figure? Be sure to count the cubes that The Parthenon; Athens, Greece are hidden from view. 31. 32. 33. 34. ART Isometric grid paper can be used to make artistic patterns and designs similar to the ones shown. Use isometric grid paper to create two different artistic designs. Color the designs. MIXED REVIEW EXERCISES Solve each system of equations graphically. (Lesson 8-2) 35. 3x y 10 36. 2x y 8 37. x 2y 7 x y 2 x 3y 10 4x 2y 22 38. 2x 2y 8 39. 3x 2y 12 40. x 3y 11 2x 2y 0 x 3y 7 x 2y 7 Show each sample space using ordered pairs. (Lesson 4-3) 41. A quarter and a dime are tossed. 42. A six-sided number cube is rolled and a dime is tossed. 43. A six-sided number cube is rolled and a spinner with five different colors is spun. (Use the numbers 1–5 for the colors on the spinner.) 44. Two spinners, one numbered 1–4 and the other lettered A–F, are spun. mathmatters2.com/self_check_quiz Lesson 10-5 Isometric Drawings 445 10-6 Orthogonal Drawings Goals I Sketch orthogonal drawings of figures. I Sketch and use foundation drawings. Applications Engineering, Interior design, History, Safety Use 11 cubes, preferably interlocking ones. D C 1. Label the four corners of a sheet of paper A, B, C, and D. 2. Build a structure by stacking the given number of cubes in 3 2 each position as shown in the figure. 1 2 3 3. Match each figure below with corner views A, B, C, or D. Try to visualize the answer in your mind; then check the A B actual structure to verify. figure 1 figure 2 figure 3 figure 4 BUILD UNDERSTANDING While perspective drawings and isometric drawings are useful to help visualize a three-dimensional figure, they may not show all the details of an object that you need. An orthogonal drawing, or orthographic drawing, gives top, front, and side views of a three-dimensional figure as seen from a “straight on” viewpoint. In an orthogonal drawing, solid lines represent any edges that show. Example 1 Make an orthogonal drawing of the figure. top Solution Draw the front view first. Draw the top view above it. Make sure it has the same width as the front view. front right side Draw the right-side view. Make sure it has the same height as the front view and the same depth as the top view. 446 Chapter 10 Three-Dimensional Geometry Example 2 ENGINEERING The isometric drawing shows part of an air- conditioning duct. a. Make an orthogonal drawing of the duct showing the front, top, and right-side views. b. The length of one cube side is 1.6 m. The building engineer needs the surface area of the top to be between 10 m2 and 10.5 m2. Is the duct within the necessary requirements? Solution a. Think of the duct as a combination of four cubes. Then make each view of the orthogonal drawing. top b. Calculate the surface area of the top of the figure. It is composed of four cube faces. 1.6 1.6 2.56 m2 Calculate the area of one cube. front right side 2 2.56 4 10.24 m Find the total surface area of the top. The duct is within the given requirements. 54 67 54 A foundation drawing shows the base of a structure and the height of each part. They are often used by architects and design engineers. The figure shows a foundation drawing of the Sears Tower in 67 98 67 Chicago. Each number represents how many stories are in each section. 41 98 41 Example 3 Create a foundation drawing for the isometric drawing. Assume the drawing is viewed from the lower left-hand corner. Solution First draw the orthogonal top view of the figure. 3 2 1 Then determine how many cubes belong in each section, and write the number to complete the 2 foundation drawing. 1 TRY THESE EXERCISES Make an orthogonal drawing labeling the front, top, and right-side views. 1. 2. 3. 4. mathmatters2.com/extra_examples Lesson 10-6 Orthogonal Drawings 447 The figure is a cube with a “half cube” on top of it. Tell which view of the figure is shown. 5. 6. 7. Create a foundation drawing for each figure. Assume the drawing is viewed from the lower left-hand corner. 8. 9. 10. PRACTICE EXERCISES PRACTICE EXERCISES • For Extra Practice, see page 618. Make an orthogonal drawing labeling the front, top, and right-side views. 11. 12. 13. For each foundation drawing, sketch the front and right orthogonal views. 14. 15. 16. 5 5 1 2 3 1 3 1 2 1 3 2 front front 3 2 1 front 17. WRITING MATH Choose an object and describe how the orthogonal drawings of the front, top, and right-side views are different or the same. 18. Make an orthogonal drawing of a regular six-sided number cube showing a view of each of the six sides. 448 Chapter 10 Three-Dimensional Geometry 19. SAFETY Most skyscrapers are built using steel girders shaped like the capital letter I. The shape avoids buckling and can support heavy loads. Make an orthogonal drawing of the girder showing the front, top and right-side views. 20. WRITING MATH Investigate the meaning of the word “orthogonal.” Why do you think it is used to describe the “straight on” view of a figure? 21. HISTORY Make an orthogonal drawing of the Sears Tower in Chicago. (Refer to the foundation drawing of the Sears Tower found earlier in this lesson.) EXTENDED PRACTICE EXERCISES Create an orthogonal drawing for each figure. 22. 23. 24. 25. CRITICAL THINKING The drawing shows top and front views of a building. Draw three possible right-side views of the building. 26. Make an orthogonal drawing showing the front, top and right-side views of a staircase that is one cube wide and reaches a height of top ten cubes. Then make a foundation drawing of the staircase. 27. INTERIOR DESIGN If the height of each cube in Exercise 26 is 8 in., how many square inches of carpeting are needed to completely cover the front and top edges of the staircase? front MIXED REVIEW EXERCISES Use grid paper. Graph each function for the domain of real numbers. (Lesson 6-6) 28. y x2 1 29. y x2 x 2 30. y x2 x 4 31. y 2x 2 3x 1 32. y 3x 2 4 33. y 2x 2 x 1 2 2 2 34. y 2x 2 35. y x 3x 4 36. y 2x x 5 Write each phrase as a variable expression. (Lesson 2-3) 37. six times a number divided by three 38. the product of seven and a number 39. two more than 12 times a number 40. the quotient of negative six and a number 41. a number less nine 42. negative three times a number mathmatters2.com/self_check_quiz Lesson 10-6 Orthogonal Drawings 449 Review and Practice Your Skills PRACTICE LESSON 10-5 Use the isometric drawing for Exercises 1–3. Assume that no cubes are hidden from view. 1. How many cubes are used in the drawing? 2. How many cube faces are exposed in the figure? 3. If the length of one of the cubes is 2.8 in., what is the total surface area of the figure to the nearest tenth of an inch? Create an isometric drawing of each figure. 4. pentagonal prism 5. figure composed of 10 cubes 6. figure with 7 faces 7. triangular prism PRACTICE LESSON 10-6 Create a foundation drawing for each isometric drawing. Assume the drawing is viewed from the lower left-hand corner. 8. 9. 10. Make an orthogonal drawing labeling the front, top and right-side views. 11. 12. 13. For each foundation drawing, sketch the front and right orthogonal views. 14. 15. 16. 3 2 3 3 1 2 1 2 2 1 2 front 1 1 1 front front 450 Chapter 10 Three-Dimensional Geometry PRACTICE LESSON 10-1–LESSON 10-6 Draw a rectangular pyramid. (Lesson 10-1) 17. How many vertices does it have? 18. How many faces does it have? 19. How many bases does it have? 20. What is the surface area of a rectangular prism with dimensions of 2 in. by 6 in. by 4 in.? (Lesson 10-3) 21. Draw a net for a pentagonal prism. (Lesson 10-2) For each foundation drawing, sketch the front and right orthogonal views. (Lesson 10-6) 22. 23. 24. 3 3 1 2 1 1 2 1 EER FEATURE #2 2 2 1 1 front front 1 front EXHIBIT DESIGNER Career – Exhibit Designer Workplace Knowhow Empire State Building E xhibit designers organize and design products and materials so they serve the intended purpose and are visually pleasing. Some exhibit Section Width* (feet) Depth* (feet) Height* (feet) designers work in museums to create, design, 1 833 375 125 install, disassemble and store historical displays. 2 667 292 250 A museum exhibit designer is designing models 3 500 208 750 of notable skyscrapers for a historical exhibit. 4 250 125 125 Accurately communicating the design ideas to * Dimensions are not actual measurements the model creator is important. The figure shown of the Empire State Building is a two-point perspective drawing of the Empire State Building (without the lightning rod) in New 4 York City, NY. Refer to the drawing and the table. 3 height 1. Create an orthogonal drawing of the Empire State Building. 2. Create an isometric drawing of the Empire State Building. 2 3. Create a one-point perspective drawing of the Empire State Building. 1 width depth mathmatters2.com/mathworks Chapter 10 Review and Practice Your Skills 451 10-7 Volume of Prisms and Pyramids Goals I Use a formula to find the volume of prisms. I Use a formula to find the volume of pyramids. Applications Earth science, Hobbies, Packaging, History, Machinery Use sheets of heavy paper, a ruler, tape and dried beans. 10 cm 1. Use the patterns to make a cube and a square pyramid. Both shapes should be missing one side. 2. Completely fill the cube with dried beans. Record the number of beans used. 3. Pour the beans directly from the cube to the pyramid. Approximately how many times can you fill the pyramid? 4. What relationship do you see? 10 cm 10 cm BUILD UNDERSTANDING Recall that volume is a measure of the number of cubic units needed to fill a region of space. To find the volume of any prism, multiply the area of the base by the height of the prism. Volume of V Bh a Prism where B is the area of the base and h is the height of the prism. Example 1 Find the volume of each prism. Problem Solving Tip a. b. 25 m 7 ft Volume is always measured in cubic units 7 ft since it is found by 7 ft 12 m 14 m multiplying an area (which is measured in square units) by a length. Solution a. The base is a rectangle. b. The base is a square. B 14 12 B lw B 72 B s2 B 168 B 49 Use the volume formula. Use the volume formula. V 168 25 V Bh V 49 7 V Bh V 4200 V 343 The volume is 4200 m3. The volume is 343 ft3. 452 Chapter 10 Three-Dimensional Geometry Example 2 14 cm 9 cm Find the volume of the triangular prism. 25 cm Solution The base is a right triangle. 1 1 B 9 14 B bh Problem Solving 2 2 B 63 Tip Use the volume formula. To find the volume of a rectangular prism, use V 63 25 V Bh the formula V lwh. V 1575 To find the volume of a cube, use the formula The volume is 1575 cm3. V s 3. Example 3 RECREATION A rectangular swimming pool is filled with 1750 ft3 of water. The pool width is 14 ft and the pool depth is 5 ft. Find the length of the pool. Solution Use the volume formula for a rectangular prism. 1750 l 14 5 V lwh 1750 70l Solve for l. 1750 70l 70 70 25 l The length of the pool is 25 ft. The volume of a prism is 3 times the volume of a pyramid. To find the volume of a pyramid, take 1 the product of the area of the base and the height. 3 1 Volume of V Bh 3 a Pyramid where B is the area of the base and h is the height. Example 4 30 m Find the volume of the rectangular pyramid. 16 m 24 m Solution Find the area of the rectangle base. Use the volume formula. 1 1 B 24 16 B lw V 384 30 V Bh 3 3 B 384 m2 V 3840 3 The volume of the pyramid is 3840 m . mathmatters2.com/extra_examples Lesson 10-7 Volume of Prisms and Pyramids 453 TRY THESE EXERCISES Find the volume of each figure. 1. 8m 2. 9 mm 3. 4. 6 in. 10.3 mm 12 in. 21 m 23 m 18 mm 8 in. 6 in. 12 in. 10 m 11 m 14 m 5. PACKAGING A gift box that is 15 in. high is a prism with a square base. It has a volume of 540 in.3. What is the length of the sides of its base? 6. Find the volume of a square pyramid if the length of a side of its base is 23 ft and its height is 65 ft. Round the answer to the nearest tenth. 7. A triangular prism has a volume of 54 cm3. The triangular base has a height of 4 cm and a base of 6 cm. What is the height of the prism? 8. WRITING MATH Given the volume of a pyramid and the area of its base, describe how you can find its height. PRACTICE EXERCISES PRACTICE EXERCISES • For Extra Practice, see page 619. Find the volume of each figure. 9. 10. 7m 11. 12. 4m 4m 3m 6m 9.5 m 8m 24.4 m 10 m 31 cm 12 m 9m 9m 13. 14. 15. 16. 15 in. 17 ft 18 ft 7.2 in. 14 m 20 m 12 in. 22 ft 10 m 6m 8m 6m 6m 17. WRITING MATH Explain the difference between the surface area and the volume of a three-dimensional figure. 18. A cereal box is 8 in. long and 4 in. wide. Its volume is 368 in.3. What is the height of the box? 19. The perimeter of the base of a square prism measures 60 cm. The height of the prism is 25 cm. What is the volume of the prism? 20. The base of a prism is an isosceles right triangle with sides measuring 5 2 cm, 5 2 cm and 10 cm. The height of the prism is 20 cm. What is the volume of the prism? 454 Chapter 10 Three-Dimensional Geometry 21. MACHINERY A dump truck has a bed that is 12 ft long and 8 ft wide. The walls of the bed are 4.5 ft high. When the truck is loaded, no material can be higher than the walls of the bed. If topsoil costs $18/yd3, what is the cost of a full truckload of topsoil? 22. EARTH SCIENCE When water freezes, its volume increases by about 10%. A tin container that measures 1 ft by 10 in. by 9 in. is exactly half full of water. It is left outside on a winter day, and all of the water freezes. What is the approximate volume of ice in the tin? 23. HISTORY The square base of an Egyptian pyramid is 62 m long. If the height of the pyramid is 78 m, what is its volume? EXTENDED PRACTICE EXERCISES 2m Find the volume of each figure. 24. 25. 17 cm 26. 2m 4 cm 5 ft 5 ft 10 m 20 m 26 cm 7 ft 21 cm 6 ft 10 m 27. HOBBIES An aquarium has a length of 15 in. and a width of 11 in. A rock put into the aquarium causes the water level to rise by 2 in. The rock is completely submerged. What is the volume of the rock? 28. Suppose you wish to make a cardboard box with a volume of 1000 cm3. What dimensions would you give to the box in order to use the least amount of cardboard? 29. CRITICAL THINKING The sides of a cube each measure 1 ft. If each side is increased by 1 in., by how many cubic inches would the volume increase? 30. CHAPTER INVESTIGATION Research a significant historical event that was happening at each date on your timeline. MIXED REVIEW EXERCISES Simplify. (Lesson 9-4) 31. 2z(4z w) 32. 3d(2d 3c) 33. 5rs(3r 4s) 34. 6xy(2x 5y) 35. 2p(4p 3r) 5p(7p 8r) 36. 3st(3s 3t) 2st(2s 5t) 37. 4yz(y 3z) 2yz(6y 4z) 38. 5cd(2c 3d) 2cd(c 5d) Copy quadrilateral KLMN on grid paper. Draw 4 y each dilation. (Lesson 7-5) K L 2 39. scale factor 2, center (0, 0) x 1 40. scale factor , center (0, 0) 4 2 2 4 2 41. scale factor 3, center L 2 2 N M 42. scale factor , center N 4 3 mathmatters2.com/self_check_quiz Lesson 10-7 Volume of Prisms and Pyramids 455 10-8 Volume of Cylinders, Cones and Spheres Goals Applications I Find the volume of cylinders, cones and spheres. Sports, Horticulture, Astronomy, History, Chemistry FITNESS A person’s vital capacity is the measure of the volume of air held in his or her lungs. Take a deep breath, and blow into a balloon as much air as possible. Trap the air by tying off the balloon. 1. Push on the end of the balloon so that it forms a sphere. Then use a tape measure to find the circumference of the balloon. Let this measure be C. 2. Find your vital capacity (VC) by using the formula C3 VC 2. 6 BUILD UNDERSTANDING Recall that the general formula for finding the volume of a prism is V Bh, r where B is the area of the base. Since the base of a cylinder is circular, replace B in this formula with r 2, the formula for the area of a circle. h 2 Volume of V r h a Cylinder where r is the radius of the base and h is the cylinder’s height. 4.5 m Example 1 Check 10 m Find the volume of the cylinder. Understanding How is the formula for Solution the volume of a cone similar to the formula for V r 2h Use the formula for the volume of a cylinder. the volume of a pyramid? 2 V 3.14 (4.5) 10 3.14 V 635.85 The volume of the cylinder is approximately 635.85 m3. The volume of a cylinder is 3 times the volume of a cone that has the same h radius and height. To find the volume of a cone, take 1 the product of the 3 area of the base and the height of the cone. r 1 Volume of V r 2h 3 a Cone where r is the radius of the base and h is the cone’s height. 456 Chapter 10 Three-Dimensional Geometry Example 2 4 cm Find the volume of the cone. 6 cm Solution 1 2 V r h Use the formula for the volume of a cone. 3 1 V 3.14 42 6 3.14 3 V 100.48 The volume of the cone is approximately 100.48 cm3. There is also a formula for the volume of a sphere. 4 Volume of V r3 3 a Sphere where r is the radius of the sphere. Example 3 ASTRONOMY The diameter of the planet Mars is approximately 6800 km. What is the volume of Mars? Solution Assume that Mars is a sphere. Mars’ radius is half of 6800 km, or 3400 km. 4 3 V r Use the formula for the volume of a sphere. 3 4 V 3.14 (3400)3 3.14 3 V 1.6455 1011 km3 The volume of Mars is approximately 165,000,000,000 km3. Mars 5m 10 m Example 4 12 m Find the volume of the figure. Solution The total volume of the figure is the sum of the cylinder’s volume and the cone’s volume. Volume of the cylinder Volume of the cone 2 1 2 V r h V r h 3 1 V 3.14 122 5 V 3.14 122 10 3 V 2260.8 V 1507.2 Total volume 2260.8 1507.2 3768 The volume of the figure is approximately 3768 m3. mathmatters2.com/extra_examples Lesson 10-8 Volume of Cylinders, Cones and Spheres 457 TRY THESE EXERCISES Find the volume of each figure. Round to the nearest whole number. 1. 2. 19 in. 3. 5 cm 4. 8 in. 24 m 11.2 m 10.4 cm 14 in. 5. A cone with a radius of 12 cm has a volume of 753.6 cm3. What is the height of the cone? 6. What is the volume of a sphere with a radius of 35 m? 7. A cubic foot of sand has a mass of 75 lb. How many pounds of sand would fit into a cylindrical container with a radius of 10 ft and a height of 5 ft? 8. What is the volume of a hemisphere with a radius r? PRACTICE EXERCISES • For Extra Practice, see page 619. Find the volume of each figure. Round to the nearest whole number. 9. 12 m 10. 17 in. 11. 12. 13 in. 16 m 10 m 9.2 m 12 m 10 m 13. A gasoline storage tank is a cylinder with a radius of 10 ft and a height of 6 ft. How many cubic feet of gasoline will the tank hold? 14. A sphere has a volume of 523 1 in.3. What is the radius of the sphere? 3 15. SPORTS Tennis balls are sold in cylindrical cans. Each can holds three tennis balls. If the volume of the can is 150.72 in.3, what is the approximate radius of a tennis ball? 16. HORTICULTURE The figure at the right is a sketch of a proposed greenhouse that is to be shaped like a hemisphere. To 50 ft the nearest cubic foot, what is the amount of space inside this greenhouse? Find the volume of each figure. Round to the nearest tenth. 17. 7 cm 18. 4 ft 19. 10 cm 4 ft 1 ft 1 ft 4 cm 4 ft 4 ft The radius of the hole is 1 ft. 16 cm 20. WRITING MATH Explain how to find the volume of a cylinder, a cone, and a sphere. Describe the similarities among the three different volume formulas. 458 Chapter 10 Three-Dimensional Geometry 21. HISTORY Early astronomers estimated the equatorial radius of the earth to be approximately 520 km. The actual radius is about 6378 km. What is the approximate volume of the earth? How inaccurate was the astronomers’ estimate of the earth’s volume? 22. A cylinder has a circumference of 12 m and a height of 4.7 m. What is the volume of the cylinder? Round the answer to the nearest whole number. 23. CHEMISTRY A chemist pours 188.4 cm3 of a liquid into a glass cylinder that has a radius of 2 cm. How many centimeters deep is the liquid? DATA FILE For Exercises 24–27, refer to the data on various balls used in sports on page 571. Find the volume of each sports ball. Round to the nearest tenth. 24. baseball 25. golf ball 26. volleyball 27. croquet ball EXTENDED PRACTICE EXERCISES CRITICAL THINKING Each of the cans shown in the diagram has a volume of 1000 cm3. (Recall that 1000 cm3 1 L.) 3 cm 4 cm 5 cm 6 cm 7 cm 1L 8 cm 1L 1L 1L 1L 1L a b c d e f 28. Find the height of each can rounded to the nearest tenth. 29. Which two cans have the least surface area? 30. If you manufacture one liter cans, what radius would you use so that you save the most material? 31. SPREADSHEET Write a spreadsheet program to calculate the volume of a sphere. Input the formula in the first cell, and calculate the volumes of spheres with radii 2 m, 4 m, 8 m and 16 m. Do you notice a pattern? Explain. MIXED REVIEW EXERCISES Find the value of the determinant of each matrix. (Lesson 8-5) 4 5 1 8 6 2 5 6 0 1 32. 33. 34. 35. 36. 3 6 2 5 4 7 12 8 2 6 3 3 8 13 4 7 4 9 11 4 37. 4 38. 39. 40. 41. 6 10 9 2 8 2 5 M 9 3 N In the figure, m MLN 35° and m NLO 55°. Find each measure. (Lesson 5-8) L _ 42. mNO - 43. m MN O - 44. mMNO - 45. m MPO P mathmatters2.com/self_check_quiz Lesson 10-8 Volume of Cylinders, Cones and Spheres 459 Review and Practice Your Skills PRACTICE LESSON 10-7 Find the volume of each figure. Round to the nearest tenth. 1. 4m 2. 4 mm 3. 12 mm 9m 15 m 8.4 mm 13 m 10 m 5m 4. 2 in. 5. 6. 7m 13 m 5m 6 in. 9m 4 in. 3 in. 8m 7 in. 6m 7. 8. 9. 5 12 cm cm 6 cm 19 m 6 cm 18 cm 6 cm 11 m 11 m 10. A triangular prism has a volume of 108 m3. The triangular base has a height of 9 m and a base of 6 m. What is the height of the prism? PRACTICE LESSON 10-8 Find the volume of each figure. Round to the nearest whole number. 11. 12. 13. 8 in. 24 m 10 m 7 in. 14 in. 14. 15. 5m 16. 6 cm 9m 11 m 10 cm 17. The volume of a cone with a 14-mm diameter is 820 mm3. Find the height. 18. The radius of an inflated beach ball is 15 in. What is the amount of air inside? 460 Chapter 10 Three-Dimensional Geometry PRACTICE LESSON 10-1–LESSON 10-8 19. Find the minimum amount of gift wrap needed to cover a box that measures 2 ft by 3 ft by 6 in. (Lesson 10-2) 20. Sketch a net for the three-dimensional figure. (Lesson 10-2) 21. Create an isometric drawing of a triangular prism. (Lesson 10-5) 22. Create an isometric drawing of a figure composed of six cubes. (Lesson 10-5) Find the surface area and volume of each figure. Round to the nearest tenth. (Lessons 10-3, 10-7 and 10-8) 23. 24. 18 cm 25. 10 m 23 cm 16 in. 8m 17 m 12 in. 6m 26. 27. 28. 7.02 ft 7.3 ft 3 in. 4 ft 3 in. 7 mm 3 in. 22 mm 8 mm 4 ft Identify each figure and state the number of faces, vertices and edges. (Lesson 10-1) 29. 30. 31. Trace each perspective drawing to locate the vanishing point(s). (Lesson 10-4) 32. 33. 34. Chapter 10 Review and Practice Your Skills 461 10-9 Problem Solving Skills: Length, Area, and Volume Formulas are mathematical tools that help guide you to a solution. The strategy, use an equation or formula, is appropriate when you have Problem Solving data that can be substituted into a formula or equation. If you do not Strategies know the formula, write an equation that states the relationships in Guess and check the problem. In either case, solve the equation to find the solution. Look for a pattern Solve a simpler problem Problem Make a table, chart or list CONSTRUCTION The tank of a raised cylindrical water tower is 26 ft Use a picture, high with a radius of 11 ft. There is a 2-ft-wide walkway around the diagram or model base of the tank. Act it out a. About how many gallons of paint are needed to paint the exterior Work backwards of the tank if 1 gal of paint will cover 300 ft2? Eliminate possibilities b. There is a railing around the walkway. How long is it? ✔ Use an equation or c. How many gallons of water does the tank hold? (The volume of formula 1 gal is approximately 0.1337 ft3.) Solve the Problem a. Each gallon of paint covers an area of 300 ft2. SA 2 r 2 2 rh Use the formula for surface area of a cylinder. 2 SA (2 3.14 11 ) (2 3.14 11 26) 3.14 SA 760 1796 2556 The surface area is approximately 2556 ft2. Since each gallon of paint covers 300 ft2, divide the surface area by 300 ft. 2556 300 8.5 It will take approximately 8.5 gal of paint. b. The length of the railing is the circumference of a circle with a radius 2 ft greater than the radius of the tank. C 2 r Use the formula for the circumference of a circle. C 2 3.14 (11 2) 3.14 C 81.6 The railing is approximately 81.6 ft long. c. The number of gallons the tank will hold is the volume of the tank. V r 2h Use the formula for the volume of a cylinder. 2 V 3.14 11 26 9878.4 3.14 The volume is approximately 9878.4 ft3. Since 1 gal is approximately 0.1337 ft3, the tank holds approximately 9878.4 0.1337 73,884.8 gal. 462 Chapter 10 Three-Dimensional Geometry TRY THESE EXERCISES Five-step Plan Round answers to the nearest tenth. 1 Read 1. A cypress tree has a radius of 18.75 ft. If it is fenced in so that there is a 2 Plan border 8 ft wide around the trunk of the tree, how many feet of fencing 3 Solve 4 Answer are needed? 5 Check 2. South African ironwood is the world’s heaviest wood, weighing about 90 lb/ft3. How much would an 8-in. cube of ironwood weigh? 3. PACKAGING Find the minimum amount of paper needed for the label of a cylindrical soup can that is 7 in. high with a radius of 2.25 in. 4. The windows of a building have a total area of 32.4 m2. Each window is a rectangle measuring 1.2 m by 1.8 m. How many windows are in the building? PRACTICE EXERCISES Round answers to the nearest tenth. 5. LANDSCAPING A pound of grass seed covers an area of 250 ft2. How many pounds of seed would you need for a rectangular lawn measuring 35 ft by 50 ft? 6. The roof of a shed is a square pyramid with sides of 2.4 m. The height of each triangular face is 2 m. How many square meters of tar paper would it take to cover the roof ? 7. A circular swimming pool has a diameter of 17 ft and a depth of 5 ft. If the pool is considered to be full when the water level is 1 ft below the rim of the pool, how many cubic feet of water does it take to fill the pool? 8. A cardboard hat is made of a cone with radius 5 in. and slant height 14 in. The circular brim is 4-in. wide. Find the minimum amount of cardboard needed to make the hat. 9. WRITING MATH Write a problem that can be solved by applying the formula for length, area, or volume. Provide an answer for your problem. 10. CRITICAL THINKING The names and sizes of wooden boards specify the dimensions before they are dried and planed. For example, a one-by-ten board is actually 3 in. by 9 1 in. If a patio is built using 15 4 4 one-by-ten boards that are each 12 ft long, what is the area of the patio? MIXED REVIEW EXERCISES Simplify. (Lesson 9-3) 14b 2 51a 2c 3 27g 3h 7m 3n 4 49k 3l 2 11. 12. 13. 14. 15. 2b 3ac 9gh m2n 14k 3l 2 Calculate each permutation. (Lesson 4-6) 16. 8P2 17. 5P2 18. 12P4 19. 10P6 20. 9P4 21. 12P8 22. 9P6 23. 10P3 24. 14P5 25. 11P3 Lesson 10-9 Problem Solving Skills: Length, Area and Volume 463 Chapter 10 Review VOCABULARY Choose the word from the list that best completes each statement. ?__ 1. A(n) ___ is a polyhedron with only one base. a. cone ?__ 2. A(n) ___ is a polyhedron with two identical parallel bases. b. cylinder ?__ 3. The ___ of a cone is the length of a segment drawn from its c. face vertex to its base along the side of the cone. d. isometric ?__ 4. A(n) ___ is the set of all points in space that are a given distance from a given point. e. orthogonal 5. A one-point perspective drawing has one ___ ?__. f. prism 6. A(n) ___ is a three-dimensional figure with two congruent ?__ g. pyramid circular bases that lie in parallel planes. h. slant height 7. The sum of the areas of all bases and faces of a three- i. sphere ?__. dimensional figure is called its ___ j. surface area 8. The number of cubic units needed to fill a three-dimensional figure is called its ___ ?__. k. vanishing point 9. A(n) ___ drawing shows three sides of a three-dimensional ?__ l. volume figure. ?__ 10. A(n) ___ drawing shows the individual sides of a three- dimensional figure. LESSON 10-1 Visualize and Represent Solids, p. 422 E F A polyhedron is a closed, three-dimensional figure made up of polygonal surfaces called faces. A segment that is the intersection of two faces is an edge. The point at which B C D three or more edges intersect is a vertex. Use the figure at the right. A 11. Identify the figure. 12. Identify its base(s). 13. Identify its lateral face(s). 14. State the number of vertices. 15. State the number of edges. 16. Identify all edges that are skew to CF. LESSON 10-2 Nets and Surface Area, p. 426 A net is a two-dimensional pattern that can be folded to form a three- dimensional figure. A net can be used to find surface area. Draw a net for each three-dimensional figure. Then find the surface area. 17. 18. 2.4 mm 19. 6 cm 5m 4.6 mm 12 m 18 m 4 cm 4 cm 464 Chapter 10 Three-Dimensional Geometry LESSON 10-3 Surface Area of Three-Dimensional Figures, p. 432 Surface areas can be found using these formulas. rectangular prism: cylinder: cone: SA 2(lw lh wh) SA 2 r2 2 rh SA rs r2 Find the surface area of each figure. Round to the nearest hundredth. 20. 2.1 m 21. 8m 22. 5 ft 6.5 m 20.2 m 11 ft 3.8 m 23. A soup can has a height of 10 cm and a diameter of 6.5 cm. Find the amount of steel needed to make the can. 24. A pet carrier is in the shape of a rectangular prism. It is 2.5 ft long, 1 ft high and 1.25 ft wide. What is the surface area of the carrier? LESSON 10-4 Perspective Drawings, p. 436 A perspective drawing is a way of drawing objects on a flat surface so that they look the same as they appear in real life. 25. Sketch a rectangular tissue box in one-point perspective. 26. Sketch a rectangular tissue box in two-point perspective. 27. Locate the vanishing point in the perspective drawing. LESSON 10-5 Isometric Drawings, p. 442 An isometric drawing shows an object from a corner view so that three sides of the object can be seen in a single drawing. Give the number of cubes used to make each figure and the number of cube faces exposed. Be sure to count the cubes that are hidden from view. 28. 29. 30. LESSON 10-6 Orthogonal Drawings, p. 446 An orthogonal drawing gives top, front and side views of a three-dimensional figure as seen from a “straight on” viewpoint. Make an orthogonal drawing labeling the front, top and right-side views. 31. 32. 33. Chapter 10 Review 465 For each foundation drawing, sketch the front and right orthogonal views. 34. 35. 36. 2 1 1 1 3 1 2 3 2 1 2 2 4 3 4 1 1 LESSON 10-7 Volume of Prisms and Pyramids, p. 452 Volume can be found using these formulas. 1 prism: V Bh pyramid: V 3 Bh 37. Find the volume of a square pyramid if the length of a side of its base is 15 cm and its height is 23 cm. 38. A rectangular prism has a base that measures 5 ft by 3 ft. Its volume is 300 ft3. What is its height? 39. A rectangular cake pan is 2 in.-by-13 in.-by-9 in. A round cake pan has a diameter of 8 in. and a height of 2 in. Which will hold more cake batter, the rectangular pan or two round pans? LESSON 10-8 Volume of Cylinders, Cones and Spheres, p. 456 Volume can be found using these formulas. 1 2 4 3 cylinder: V r2h cone: V 3 rh sphere: V 3 r Find the volume of each figure. Round to the nearest tenth. 40. cylinder: r 6.1 mm, h 3.8 mm 41. cone: r 4.2 cm, h 11 cm 42. sphere: r 6 mm 43. cone: r 5 in., h 15 in. LESSON 10-9 Problem Solving Skills: Length, Area and Volume, p. 462 The strategy, use an equation or formula, is appropriate when you have data that can be substituted into a formula or equation. 44. A rectangular room measures 12 yd by 22 ft. How many square yards of carpet will it take to carpet the entire room? 45. A rectangular room is 12 ft-by-21 ft. The walls are 8 ft tall. Paint is sold in 1-gal containers. If a gallon of paint covers 450 ft2, how many gallons of paint should be purchased to paint the walls of the room? CHAPTER INVESTIGATION EXTENSION Write a report about how the study of mathematics influences history. Your report could answer some of the following questions. How did cultural and historical events affect mathematics? How does geography affect the spread of information and new discoveries? How did mathematics help trigger major cultural events like the Industrial Revolution and the Information Age? Have all mathematicians been formally educated? How did computers change the way mathematicians work? 466 Chapter 10 Three-Dimensional Geometry Chapter 10 Assessment Use the figure to name the following. E 1. a pair of intersecting edges B F D 2. a pair of parallel edges 3. a pair of parallel faces A C 4. a pair of edges that are skew 5. the bases Draw a net for each figure. Then identify the figure and find its surface area. 6. 7. 8. 17 m 21 ft 16 cm 3 ft 8m 8m 10 ft 9 cm 9. Find the surface area of a cone with a radius of 3.1 cm and a slant height of 12.4 cm. 10. Find the surface area of the figure. 11. Locate the vanishing point of the figure. 2.5 cm 12. Create an isometric drawing of a figure composed of 5 cubes. 13. Make an orthogonal drawing of the figure you drew in Exercise 12. Show the front, top and right-side views. Find the volume of each figure. Round to the nearest tenth if necessary. 14. 15. 16. 9m 5 in. 16 m 13 in. 18 in. 10 m 4.5 m 6.2 m 4m 17. To find the number of square feet of wrapping paper needed to cover a box shaped like a rectangular prism, which formula should you use? a. P 2l 2s b. V lwh c. SA 2(lw lh wh) 18. Write a problem that can be solved by applying the formula for the volume of a sphere. Provide an answer for your problem. mathmatters2.com/chapter_assessment Chapter 10 Assessment 467 Standardized Test Practice Part 1 Multiple Choice 6. What is the slope of the line that passes through A( 3, 2) and B(5, 4)? Record your answers on the answer sheet (Lesson 6-2) provided by your teacher or on a sheet of paper. A 4 B 3 3 4 1. Desiree’s test scores are 83, 75, 86, and 82. 3 4 C D 4 3 If her teacher uses the mean, what score does she need on the fifth test in order to 7. Which line is not parallel to 2x 3y 5? have an average of 85? (Lesson 1-2) (Lesson 8-1) A 85 B 90 A 2x 3y 1 C 99 D 105 B 2x 3y 2 C 4x 6y 5 2. Of the people who buy raffle tickets, 500 win nothing, 1 wins $25, and 1 wins $100,000. If D 3x 2y 5 you were promoting the raffle and wanted to give a misleading statistic about the average 8. If 2x y 3 and x y 1, what is the value winning, which measure of central tendency of y? (Lesson 8-4) would you use? (Lesson 1-7) A 2 B 1 A mean C 1 D 2 B median C mode 9. Factor x2 4x 4. (Lesson 9-8) 2 D range A (x 2) B (x 2)2 C x(x 4) D (x 2)(x 2) 3. Which expression is not represented by 2x 1? (Lesson 2-3) 10. How many faces does a pentagonal pyramid A twice a number minus one have? (Lesson 10-1) B twice a number less than one A 5 B 6 C two times a number decreased by one C 7 D 8 D two times a number minus one 11. What is the volume of the 4. Which graph represents the solution of rectangular pyramid? (Lesson 10-7) 11 ft 3t 5 2? (Lesson 3-7) A A 176 ft3 3 2 1 0 1 2 3 B 404 ft3 6 ft B C 528 ft3 8 ft 3 2 1 0 1 2 3 D 576 ft3 C 3 2 1 0 1 2 3 D Test-Taking Tip 3 2 1 0 1 2 3 Question 11 Most standardized tests include any commonly used formulas 5. What is the measure of each angle of a regular at the front of the test booklet, but it will save you time to polygon that has 9 sides? (Lesson 5-7) memorize many of these formulas. For example, you should A 126° B 140° memorize that the volume of a pyramid is one-third the area of the base times the height of the pyramid. C 150° D 180° 468 Chapter 10 Three-Dimensional Geometry Preparing for the Standardized Tests For test-taking strategies and more practice, see pages 627–644. Part 2 Short Response/Grid In 21. The perimeter of the rectangle is 16a 2b. Write an expression for the length of the Record your answers on the answer sheet rectangle. (Lessons 9-3 and 9-4) provided by your teacher or on a sheet of paper. 2 12. What is the value of t for t 5? 5a – b (Lesson 2-8) 13. The area of a triangle can be determined by 1 22. A bowling league has n teams. You can use using the formula A 2 bh. Solve this formula 1 1 the expression n2 n to find the total 2 2 for h. (Lesson 3-2) number of games that will be played if each team plays each other team exactly once. 14. Solve 2(3t 6) t 8. (Lesson 3-4) Factor this expression. (Lessons 9-7) 15. A bag contains 4 red marbles, 3 blue marbles, 23. What is the surface area of a cube with sides and 2 white marbles. One marble is chosen of 7 in.? (Lesson 10-3) without replacement. Then another marble is chosen. What is the probability that the first marble is red and the second marble is blue? 24. A can of soup is 12 cm high and has a diameter (Lesson 4-5) of 8 cm. A rectangular label is being designed for this can of soup. If the label will cover the surface of the can except for its top and 16. Find the value of x in circle Q. (Lesson 5-8) bottom, what is the width and length of the x° label, to the nearest centimeter? (Lesson 10-9) Q Part 3 Extended Response Record your answers on a sheet of paper. 17. The center of a circle is located at the origin Show your work. of a coordinate plane. If A(5, 12) is on the circle, what is the radius of the circle? 25. Describe the three-dimensional figure. (Lesson 6-1) Include the number of cubes needed to make the figure and the number of cube 18. The distance a vehicle travels at a given speed faces exposed. Draw an orthogonal drawing is a direct variation of the time it travels. If a labeling the front, top, and right-side views. vehicle travels 30 mi in 45 min, how far can it Then, draw a foundation drawing. travel in 2 h? (Lesson 6-8) (Lessons 10-5 and 10-6) 19. Triangle RST with vertices R(5, 4), S(3, 1), and T(0, 2) is translated so that R' is at (3, 1). What are the coordinates of S' ? (Lesson 7-1) 20. Solve the system of equations. 26. Draw a cylinder and a cone that have the (Lesson 8-3) same volume. Explain your response and y 3x calculate the volume and surface area of each x 2y 21 figure. (Lesson 10-8) mathmatters2.com/standardized_test Chapter 10 Standardized Test Practice 469

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