Classify solids. Find volumes of prisms and cylinders.

Transcription

1 11.4 Volumes of Prisms and Cylinders Essential Question How can you find te volume of a prism or cylinder tat is not a rigt prism or rigt cylinder? Recall tat te volume V of a rigt prism or a rigt cylinder is equal to te product of te area of a base and te eigt. rigt prisms rigt cylinder V = Finding Volume Work wit a partner. Consider a stack of square papers tat is in te form of a rigt prism. ATTENDING TO PRECISION To be proficient in mat, you need to communicate precisely to oters. a. Wat is te volume of te prism? b. Wen you twist te stack of papers, as sown at te rigt, do you cange te volume? Explain your reasoning. 8 in. c. Write a carefully worded conjecture tat describes te conclusion you reaced in part (b). d. Use your conjecture to find te volume of te twisted stack of papers. 2 in. 2 in. Finding Volume Work wit a partner. Use te conjecture you wrote in Exploration 1 to find te volume of te cylinder. a. 2 in. b. 5 cm 3 in. 15 cm Communicate Your Answer 3. How can you find te volume of a prism or cylinder tat is not a rigt prism or rigt cylinder? 4. In Exploration 1, would te conjecture you wrote cange if te papers in eac stack were not squares? Explain your reasoning. Section 11.4 int_mat2_pe_1104.indd 665 Volumes of Prisms and Cylinders 665

2 11.4 Lesson Wat You Will Learn Core Vocabulary polyedron, p. 666 face, p. 666 edge, p. 666 vertex, p. 666 volume, p. 667 Cavalieri s Principle, p. 667 similar solids, p. 669 Previous solid prism pyramid cylinder cone spere base composite solid Classify solids. Find volumes of prisms and cylinders. Classifying Solids A tree-dimensional figure, or solid, is bounded by flat or curved surfaces tat enclose a single region of space. A polyedron is a solid tat is bounded by polygons, called faces. An edge of a polyedron is a line segment formed by te intersection of two faces. A vertex of a polyedron is a point were tree or more edges meet. Te plural of polyedron is polyedra or polyedrons. Core Concept Types of Solids Polyedra vertex Not Polyedra edge face prism cylinder cone pyramid spere Pentagonal prism ases are pentagons. To name a prism or a pyramid, use te sape of te base. Te two bases of a prism are congruent polygons in parallel planes. For example, te bases of a pentagonal prism are pentagons. Te base of a pyramid is a polygon. For example, te base of a triangular pyramid is a triangle. Triangular pyramid ase is a triangle. Classifying Solids Tell weter eac solid is a polyedron. If it is, name te polyedron. a. b. c. a. Te solid is formed by polygons, so it is a polyedron. Te two bases are congruent rectangles, so it is a rectangular prism. b. Te solid is formed by polygons, so it is a polyedron. Te base is a exagon, so it is a exagonal pyramid. c. Te cone as a curved surface, so it is not a polyedron. 666 Capter 11 Circumference, Area, and Volume int_mat2_pe_1104.indd 666

3 Monitoring Progress Help in Englis and Spanis at igideasmat.com Tell weter te solid is a polyedron. If it is, name te polyedron Finding Volumes of Prisms and Cylinders Te volume of a solid is te number of cubic units contained in its interior. Volume is measured in cubic units, suc as cubic centimeters (cm 3 ). Cavalieri s Principle, named after onaventura Cavalieri ( ), states tat if two solids ave te same eigt and te same cross-sectional area at every level, ten tey ave te same volume. Te prisms below ave equal eigts and equal cross-sectional areas at every level. y Cavalieri s Principle, te prisms ave te same volume. Core Concept Volume of a Prism Te volume V of a prism is V = were is te area of a base and is te eigt. Find te volume of eac prism. Finding Volumes of Prisms a. 4 cm 3 cm b. 3 cm 14 cm 2 cm 5 cm 6 cm a. Te area of a base is = 1 2 (3)(4) = 6 cm2 and te eigt is = 2 cm. V = = 6(2) = 12 Te volume is 12 cubic centimeters. b. Te area of a base is = 1 2 (3)(6 + 14) = 30 cm2 and te eigt is = 5 cm. V = = 30(5) = 150 Te volume is 150 cubic centimeters. Section 11.4 Volumes of Prisms and Cylinders 667 int_mat2_pe_1104.indd 667

4 Consider a cylinder wit eigt and base radius r and a rectangular prism wit te same eigt tat as a square base wit sides of lengt r π. r π r π r Te cylinder and te prism ave te same cross-sectional area, πr 2, at every level and te same eigt. y Cavalieri s Principle, te prism and te cylinder ave te same volume. Te volume of te prism is V = = πr 2, so te volume of te cylinder is also V = = πr 2. Core Concept Volume of a Cylinder Te volume V of a cylinder is V = = πr 2 were is te area of a base, is te eigt, and r is te radius of a base. r r Finding Volumes of Cylinders Find te volume of eac cylinder. a. 9 ft 6 ft b. 4 cm 7 cm a. Te dimensions of te cylinder are r = 9 ft and = 6 ft. V = πr 2 = π(9) 2 (6) = 486π Te volume is 486π, or about cubic feet. b. Te dimensions of te cylinder are r = 4 cm and = 7 cm. V = πr 2 = π(4) 2 (7) = 112π Te volume is 112π, or about cubic centimeters. Monitoring Progress Find te volume of te solid m 5 m 8 m Help in Englis and Spanis at igideasmat.com 5. 8 ft 14 ft 668 Capter 11 Circumference, Area, and Volume int_mat2_pe_1104.indd 668

5 Core Concept Similar Solids Two solids of te same type wit equal ratios of corresponding linear measures, suc as eigts or radii, are called similar solids. Te ratio of te corresponding linear measures of two similar solids is called te scale factor. If two similar solids ave a scale factor of k, ten te ratio of teir volumes is equal to k 3. Finding te Volume of a Similar Solid COMMON ERROR e sure to write te ratio of te volumes in te same order you wrote te ratio of te radii. Prism C V = 1536 m 3 Prism D 12 m Cylinder A and cylinder are similar. Find te volume of cylinder. Radius of cylinder Te scale factor is k = Radius of cylinder A = 6 3 = 2. Use te scale factor to find te volume of cylinder. Volume of cylinder Volume of cylinder A = k3 Te ratio of te volumes is k 3. Volume of cylinder = 2 45π Substitute. Volume of cylinder = 360π Solve for volume of cylinder. Te volume of cylinder is 360π cubic centimeters. Monitoring Progress Cylinder A 3 cm V = 45 π cm 3 Help in Englis and Spanis at igideasmat.com 6. Prism C and prism D are similar. Find te volume of prism D. Cylinder 6 cm 3 m Finding te Volume of a Composite Solid 0.39 ft Find te volume of te concrete block ft 0.33 ft To find te area of te base, subtract two times te area of te small rectangle from te large rectangle ft 0.66 ft 0.66 ft 3 ft 10 ft 6 ft = Area of large rectangle 2 Area of small rectangle = 1.31(0.66) 2(0.33)(0.39) = Using te formula for te volume of a prism, te volume is V = = (0.66) Te volume is about 0.40 cubic foot. Monitoring Progress 7. Find te volume of te composite solid. Help in Englis and Spanis at igideasmat.com Section 11.4 Volumes of Prisms and Cylinders 669 int_mat2_pe_1104.indd 669

6 11.4 Exercises Dynamic Solutions available at igideasmat.com Vocabulary and Core Concept Ceck 1. VOCAULARY In wat type of units is te volume of a solid measured? 2. WHICH ONE DOESN T ELONG? Wic solid does not belong wit te oter tree? Explain your reasoning. Monitoring Progress and Modeling wit Matematics In Exercises 3 6, matc te polyedron wit its name In Exercises 11 14, find te volume of te prism. (See Example 2.) cm cm 2.3 cm 2 cm in. 10 in. 4 m 1.5 m 2 m 5 in. 14 m A. triangular prism. rectangular pyramid C. exagonal pyramid D. pentagonal prism In Exercises 7 10, tell weter te solid is a polyedron. If it is, name te polyedron. (See Example 1.) m 11 m In Exercises 15 18, find te volume of te cylinder. (See Example 3.) ft cm 10.2 ft 9.8 cm ft m 8 ft 18 m Capter 11 Circumference, Area, and Volume int_mat2_pe_1104.indd 670

7 In Exercises 19 and 20, make a sketc of te solid and find its volume. Round your answer to te nearest undredt. 19. A prism as a eigt of 11.2 centimeters and an equilateral triangle for a base, were eac base edge is 8 centimeters. 20. A pentagonal prism as a eigt of 9 feet and eac base edge is 3 feet. In Exercises 29 and 30, te solids are similar. Find te volume of solid. (See Example 4.) 29. Prism A V = 2673 cm 3 9 cm 21. ERROR ANALYSIS Describe and correct te error in identifying te solid. Te solid is a rectangular pyramid. 30. Prism Cylinder A 12 in. 3 cm Cylinder 15 in. 22. ERROR ANALYSIS Describe and correct te error in finding te volume of te cylinder. V = 4608 π in. 3 4 ft 3 ft V = 2πr = 2π(4)(3) = 24π So, te volume of te cylinder is 24π cubic feet. In Exercises 23 28, find te missing dimension of te prism or cylinder. In Exercises 31 34, find te volume of te composite solid. (See Example 5.) ft 10 ft 2 ft 3 ft 2 ft 6 ft in. 4 in. 4 in. 23. Volume = 560 ft Volume = 2700 yd in. 8 in ft u v 11 in. 5 ft 8 ft 15 yd 7 ft 12 yd 25. Volume = 80 cm Volume = in. 3 8 cm x w 5 cm 2 in. 4 ft 2 ft 35. MODELING WITH MATHEMATICS Te Great lue Hole is a cylindrical trenc located off te coast of elize. It is approximately 1000 feet wide and 400 feet deep. About ow many gallons of water does te Great lue Hole contain? (1 ft gallons) 27. Volume = 3000 ft Volume = m ft z y 15 m Section 11.4 Volumes of Prisms and Cylinders 671 int_mat2_pe_1104.indd 671 1/30/15 1:21 PM

8 36. COMPARING METHODS Te Volume Addition Postulate states tat te volume of a solid is te sum of te volumes of all its nonoverlapping parts. Use tis postulate to find te volume of te block of concrete in Example 5 by subtracting te volume of eac ole from te volume of te large rectangular prism. Wic metod do you prefer? Explain your reasoning. 37. WRITING ot of te figures sown are made up of te same number of congruent rectangles. Explain ow Cavalieri s Principle can be adapted to compare te areas of tese figures. 41. MAKING AN ARGUMENT A prism and a cylinder ave te same eigt and different cross-sectional areas. Your friend claims tat te two solids ave te same volume by Cavalieri s Principle. Is your friend correct? Explain your reasoning. 42. THOUGHT PROVOKING Cavalieri s Principle states tat te two solids sown below ave te same volume. Do tey also ave te same surface area? Explain your reasoning. 38. HOW DO YOU SEE IT? Eac stack of memo papers contains 500 equally-sized seets of paper. Compare teir volumes. Explain your reasoning. 43. PROLEM SOLVING A barn is in te sape of a pentagonal prism wit te dimensions sown. Te volume of te barn is 9072 cubic feet. Find te dimensions of eac alf of te roof. x ft Not drawn to scale 8 ft 8 ft 18 ft 36 ft 39. OPEN-ENDED Sketc two rectangular prisms tat ave volumes of 100 cubic inces but different surface areas. Include dimensions in your sketces. 44. PROLEM SOLVING A wooden box is in te sape of a regular pentagonal prism. Te sides, top, and bottom of te box are 1 centimeter tick. Approximate te volume of wood used to construct te box. Round your answer to te nearest tent. 40. MAKING AN ARGUMENT Your friend says tat te polyedron sown is a triangular prism. Your cousin says tat it is a triangular pyramid. Wo is correct? Explain your reasoning. 4 cm 6 cm Maintaining Matematical Proficiency Reviewing wat you learned in previous grades and lessons Find te surface area of te regular pyramid. (Skills Review Handbook) m cm in. 2 m 8 cm Area of base is cm in in. 672 Capter 11 Circumference, Area, and Volume int_mat2_pe_1104.indd 672 1/30/15 1:21 PM