New Vocabulary volume

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1 -. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding Volume of a Composite Figure - Wat You ll Learn To find te volume of a prism To find te volume of a cylinder... And Wy To estimate te volume of a backpack, as in Example Volumes of Prisms and Cylinders Ceck Skills You ll Need GO for Help Lessons -9 and 0- Find te area of eac figure. For answers tat are not wole numbers, round to te nearest tent.. a square wit side lengt 7 cm 9 cm. a circle wit diameter 76.7 in. 3. a circle wit radius 0 mm 3. mm. a rectangle wit lengt 3 ft and widt ft 3 ft 5. a rectangle wit base in. and eigt in. 6. a triangle wit base cm and eigt 5 cm 7.5 cm 7. an equilateral triangle tat is on eac side 7.7 in. Mat Background New Vocabulary volume composite space figure Integral calculus considers te area under a curve, wic leads to computation of volumes of solids of revolution. Cavalieri s Principle is a forerunner of ideas formalized by Newton and Leibniz in calculus. More Mat Background: p. 596D Lesson Planning and Resources See p. 596E for a list of te resources tat support tis lesson. Bell Ringer Practice Ceck Skills You ll Need For intervention, direct students to: Areas of Rectangles and Circles Lesson -9: Examples, 5 Extra Skills, Word Problems, Proof Practice, C. Area of a Triangle Lesson 0-: Example 3 Extra Skills, Word Problems, Proof Practice, C. 0 Finding Volume of a Prism 6 Capter Surface Area and Volume Hands-On Activity: Finding Volume Explore te volume of a prism wit unit cubes. Make a one-layer rectangular prism tat is cubes long and cubes wide. Te prism will be units by units by unit.. How many cubes are in te prism? 8 cubes. Add a second layer to your prism to make a prism units by units by units. How many cubes are in tis prism? 6 cubes 3. Add a tird layer to your prism to make a prism units by units by 3 units. How many cubes are in tis prism? cubes. How many cubes would be in te prism if you added two additional layers of cubes for a total of 5 layers? 0 cubes 5. How many cubes would be in te prism if tere were 0 layers? 80 cubes Volume is te space tat a figure occupies. It is measured in cubic units suc as cubic inces (in. 3 ), cubic feet (ft 3 ), or cubic centimeters (cm 3 ). Te volume of a cube is te cube of te lengt of its edge, or V = e 3. e e e 6 Special Needs L In Example, some students may ave trouble identifying te eigt because it is not vertical. Use a drawing at te board to sow tat te eigt of a prism is te perpendicular distance between te bases. learning style: visual Below Level L Before students work troug Example, ave tem draw and label te cylinder used for te top of te backpack. Tis will clarify te formula in Step 3. learning style: visual

2 Bot stacks of paper below contain te same number of seets.. Teac Guided Instruction Te first stack forms a rigt prism. Te second forms an oblique prism. Te stacks ave te same eigt. Te area of every cross section parallel to a base is te area of one seet of paper. Te stacks ave te same volume. Tese stacks illustrate te following principle. Key Concepts Teorem -5 Cavalieri s Principle If two space figures ave te same eigt and te same cross-sectional area at every level, ten tey ave te same volume. Te area of eac saded cross section below is. Since te prisms ave te same eigt, teir volumes must be te same by Cavalieri s Principle. cm cm cm 3 cm 3 cm Hands-On Activity If you do not ave enoug cubes for eac student, demonstrate te investigation, or ave students use te isometric drawing tecniques tat tey learned in Lesson - to simulate te activity. Visual Learners Illustrate a cross section parallel to a base as you discuss Cavalieri s Principle by removing a seet from a stack of paper. Error Prevention Students may ave trouble identifying te eigt of a prism wen its base is not orizontal. Remind tem tat eigt is te measure of an altitude perpendicular to a base. You can find te volume of a rigt prism by multiplying te area of te base by te eigt. Cavalieri s Principle lets you extend tis idea to any prism. Additional Examples Find te volume of te prism. Key Concepts Teorem -6 Volume of a Prism Te volume of a prism is te product of te area of a base and te eigt of te prism. V = B Finding Volume of a Rectangular Prism B 3 in. 7 3 Find te volume of te prism. Find te volume of te prism at te rigt. 9 m For: Prism, Cylinder Activity Use: Interactive Textbook, - V = B Use te formula for volume. 0 cm = 80? 0 B? 0 80 cm 0 cm = 800 Simplify. cm Te volume of te rectangular prism is 800 cm 3. Critical Tinking Suppose te prism in Example is turned so tat te base is 0 cm by 0 cm and te eigt is cm. Explain wy te volume does not cange. Answers may vary. Sample: Multiplication is commutative. 0 m 800 m 3 0 m Lesson - Volumes of Prisms and Cylinders 65 Advanced Learners L Have students investigate ow doubling te radius, diameter, or eigt affects te volume of a cylinder. Te volume increases by a factor of, 6, or. learning style: verbal Englis Language Learners ELL Use a stack of index cards or coins to explain te term cross section and to illustrate Cavalieri s Principle. Te coin model will elp students see tat tis principle applies to cylinders as well as to prisms. learning style: visual 65

3 Guided Instruction Auditory Learners Have students explain aloud wy te formula for te volume of a prism is similar to te formula for te volume of a cylinder. Mat Tip Ask: Wy is te eigt of te prism in.? Te backpack s top is alf of a cylinder wit diameter in., so te radius of te base is Te eigt of te prism is 7 in. in. Additional Examples 3 Find te volume of te cylinder. Leave your answer in terms of p. A B C D E A B C D E 3 A B C D E A B C D E 5 A B C D E B C D E Test-Taking Tip A volume question often requires you to find a base of a solid. A base does not ave to be at te bottom (or top) of te solid. Finding Volume of a Triangular Prism Multiple Coice Find te approximate volume of te triangular prism at te rigt in Eac base of te triangular prism is an equilateral triangle. An altitude of te triangle divides it into two triangles. Te area of te base is? 8?!3, or 6!3 in.. V = B Use te formula for te volume of a prism. = 6!3? 0 Substitute. = 60!3 Simplify. = Use a calculator. Te volume of te triangular prism is about 77 in. 3. Te answer is B. Find te volume of te triangular prism at te rigt. 50 m 3 0 m 6 m 0 in m 3 in. 9 ft Finding Volume of a Cylinder 576π ft 3 To find te volume of a cylinder, you use te same formula V = B tat you use to find te volume of a prism. Now, owever, B is te area of te circle, so you use te formula B = pr to find its value. Find te volume of te composite space figure. Key Concepts Teorem -7 Volume of a Cylinder 0 cm 600 cm 3 Resources Daily Notetaking Guide - L3 Daily Notetaking Guide - Adapted Instruction L Closure cm cm 5 cm cm 0 cm Ask students to solve te following exercise. A cube wit 0-in. edges contains a cylinder 0 in. ig. Te cylinder s lateral surface touces four faces of te cube. Find te volume of te space between te cube and te cylinder to te nearest wole number. 3 nline Visit: PHScool.com Web Code: aue Te volume of a cylinder is te product of te area of te base and te eigt of te cylinder. V = B, or V = pr 66 Capter Surface Area and Volume Finding Volume of a Cylinder Find te volume of te cylinder at te rigt. Leave your answer in terms of p. V = pr Use te formula for te volume of a cylinder. = p(3) (8) Substitute. = p(7) Simplify. Te volume of te cylinder is 7p cm 3. Te cylinder at te rigt is oblique. a. Find its volume in terms of p. 56π m 3 b. Find its volume to te nearest tent of a cubic meter. 80. m 3 B 6 m 8 m r 3 cm 8 cm 66

4 7 in. in. in. A composite space figure is a tree-dimensional figure tat is te combination of two or more simpler figures. A space probe, for example, migt begin as a composite figure a cylindrical rocket engine in combination wit a nose cone. You can find te volume of a composite space figure by adding te volumes of te figures tat are combined. Finding Volume of a Composite Figure Estimation Use a composite space figure to estimate te volume of te backpack sown at te left. Step : You can use a prism and alf of a cylinder to approximate te sape, and terefore te volume, of te backpack. in. in. Step : Volume of te prism = B = (? ) = 58 Step 3: Volume of te alf cylinder = (pr ) = p(6) () = p(36)() < 6 Step : Sum of te two volumes = = 75 Te approximate volume of te backpack is 7 3. Find te volume of te composite space figure. in. 3 in. in. EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. Practice and Problem Solving in. in. in. in. in. 3. Practice Assignment Guide A B -8,, 6, 8-,, 5, 9, 36 A B 9-3, 5, 7,, 6-8, C Callenge 37-0 Test Prep -5 Mixed Review 6-5 Homework To ceck students understanding of key skills and concepts, go over Exercises 0,, 8,, 9. Connection to Algebra Exercises Use tese exercises to assess weter students substitute correctly for variables. Alternative Metod Exercise Tis figure is a prism wose vertical bases are a combination of sapes. Ask: Wic letter best describes te sape of te base? L Have students use te area of tis base to find te volume of te prism. A GO Practice by Example for Help Example (page 65) Example (page 66) In Exercises 8, find te volume of eac prism. 80 m m in. 3 m 80 in. 0 m. Te base is a square, cm on a side. Te eigt is 3.5 cm. cm ft mm 70 mm 3 8 cm 3 ft 5 ft 0 mm about mm 8. Te base is a triangle wit a leg of Te eigt is.. 3 GPS Guided Problem Solving L3 Enricment Reteacing Adapted Practice L L L Practice Name Class Date Practice - Angle Measures and Segment Lengts Find te value of x Algebra Find te value of eac variable using te given cords, secants, and tangents. If your answer is not a wole number, round it to te nearest tent. L3 Lesson - Volumes of Prisms and Cylinders z 60 z 0 Pearson Education, Inc. All rigts reserved z x 9 x.5 y z y x 0 y

5 Exercise Discuss wy te weigts of fluids and gases are given per unit of volume. Example 3 (page 66) Find te volume of eac cylinder in terms of π and to te nearest tent cm. 5 m Exercise 6 Remind students tat polygons wit equal areas need not ave equal perimeters. Similarly, space figures wit equal volumes need not ave equal surface areas. Connection to Ecology Exercise Have students investigate ow plants can improve te quality of indoor air. Example (page 67) 88π in. 3, π cm3, 5.7 cm π m 3, 7.8 m 3 Find te volume of eac composite space figure to te nearest wole number.. cm cm cm cm 0 in. cm in. 8 cm 0 cm in. 6 m 3 3 Error Prevention! Exercise 3 Remind students wo multiply by to convert cubic feet to cubic inces tat ft 3 means ft? ft? ft, so ft 3 = in.? in.? in. or 7 3 Exercise 8 Some students may incorrectly substitute te 9-in. diameter instead of te.5-in. radius in V 5 pr. B Apply Your Skills. a. Wat is te volume of a waterbed mattress tat is 7 ft by ft by ft? 8 ft 3 b. To te nearest pound, wat is te weigt of te water in a full mattress? (Water weigs 6. lb/ft 3.) 77 lb 5. Find te volume of te lunc box sown at te rigt 3 in. to te nearest cubic inc. 50 in Open-Ended Give te dimensions of two rectangular prisms tat ave volumes of 80 cm 3 eac but also ave 0 in. different surface areas. Answers may vary. Sample: cm by cm by 0 cm; cm by cm by 5 cm Find te eigt of eac figure wit te given volume cm 6 3 ft 9 cm V = 3p cm 3 V = 7 ft 3 V = 3 0. Ecology Te isolation cube at te left measures 7 in. on eac side. Wat is its volume in cubic feet? 9,683 ft 3. Environmental Engineering A scientist suggests keeping indoor air relatively clean as follows: Provide two or tree pots of flowers for every 00 square feet of floor space under a ceiling of 8 feet. If your classroom as an 8-ft ceiling and measures 35 ft by 0 ft, ow many pots of flowers sould it ave? 8 pots. Find te volume of te oblique prism pictured at te rigt Tank Capacity Te main tank at an aquarium is a Real-World Connection cylinder wit diameter 03 ft and eigt 5 ft. 809,37 ft 3 Careers An ecologist studies a. Find te volume of te tank to te nearest cubic foot. living organisms and teir ft environments. b. Convert your answer to part (a) to cubic inces.,398,88,73 3 c. If gallon < 3 in. 3, about ow many gallons does te tank old? 6,05,765 gal. Writing Te figures at te rigt can be covered by equal numbers GO nline of straws tat are te same lengt. Homework Help Describe ow Cavalieri s Principle Visit: PHScool.com could be adapted to compare te Web Code: aue-0 areas of tese figures. Answers may vary. Sample: If two plane figures ave te same eigt and te same widt at every level, ten tey ave te same area. 68 Capter Surface Area and Volume 68

6 Problem Solving Hint In Exercise 5, find te lengt, widt, and eigt along te axes. 9. Bulk; cost of bags $67.50, cost of bulk is N$ cylinder wit r and ; 6π units 3 3. cylinder wit r and ; 3π units 3 3. cylinder wit r and ; 6π units cylinder wit r 5,, and a ole of radius ; 8π units 3 5. Coordinate Geometry Find te volume of te rectangular prism at te rigt. 80 units 3 6. Te volume of a cylinder is 600p cm 3. Te radius of a base of te cylinder is 5 cm. Wat is te eigt of te cylinder? cm 7. Te volume of a cylinder is 35p cm 3.Te eigt of te cylinder is 5 cm. Wat is te radius of a base of te cylinder? 3 cm 8. Multiple Coice A cylindrical water tank as a diameter of 9 inces and a eigt of inces. Te water surface is.5 inces from te top. About ow muc water is in te tank? A 60 in in x 9. Landscaping To landscape er 70 ft-by-60 ft rectangular backyard, Joy is GPS planning first to put down a -in. layer of topsoil. Se can buy bags of topsoil at $.50 per 3-ft 3 bag, wit free delivery. Or, se can buy bulk topsoil for $.00/yd 3, plus a $0 delivery fee. Wic option is less expensive? Explain. See left. Visualization Te plane region is revolved completely y about te given line to sweep out a solid of revolution. 3 Describe te solid and find its volume in terms of π See left. 30. te x-axis 3. te y-axis O 3 x 3. te line y = 33. te line x = 5 A cylinder as been cut out of eac solid. Find te volume of te remaining solid. Round your answer to te nearest tent in. cm O in. z 5 y 9 in.. Assess & Reteac Lesson Quiz Find te volume of eac figure to te nearest wole number ft 3 3 m 0 ft 9 m 8 ft 0 ft 3 in. 37a. circumference 8 in. and eigt in.: V N 63.π in. 3 ; circumference in. and eigt 8 in.: V N 8.8π in. 3 ; one is about 0.8 times te volume of te oter. b. about 6. by 3.0 in. C Callenge 5 cm 5.7 cm A closed box is 9 in. by in. by 0. 3 on te inside and in. by by 7 in. on te outside. Find eac measurement. 7 in. a. te outside surface area 730 in. in. b. te inside surface area 5 9 in. c. te inside volume 75 3 in. d. te volume of te material needed to make te box Any rectangular seet of paper can be rolled into a rigt cylinder in two ways. a. Use ordinary seets of paper to model te two cylinders. Compute te volume of eac cylinder. How do tey compare? a b. See left. b. Of all seets of paper wit perimeter 39 in., wic size can be rolled into a rigt cylinder wit greatest volume? (Hint: See Activity Lab, page 66.) m 3 5 m m 63 m 3 7 mm mm 6 mm 6 mm mm 800 mm 3 mm mm lesson quiz, PHScool.com, Web Code: aua-0 Lesson - Volumes of Prisms and Cylinders 69 69

7 Alternative Assessment Have eac student bring in one cylindrical food container and one saped like a prism. Distribute one cylinder and one prism to eac student, and ave tem calculate te volume of eac container and explain teir calculations. 38. Te outside diameter of a pipe is 5 cm. Te inside diameter is cm. Te pipe is m long. Wat is te volume of te material used for tis lengt of pipe? Round your answer to te nearest cubic centimeter. 87 cm A cube as a volume of M cubic units and a total surface area of 3M square units. Find te lengt of an edge of te cube. units 0. Te radius of cylinder B is twice te radius of cylinder A. Te eigt of cylinder B is alf te eigt of cylinder A. Compare teir volumes. Te volume of B is twice te volume of A. Test Prep Resources For additional practice wit a variety of test item formats: Standardized Test Prep, p. 657 Test-Taking Strategies, p. 65 Test-Taking Strategies wit Transparencies Test Prep Multiple Coice Sort Response. Wat is te volume of a rectangular prism wose edges measure ft, ft, and 3 ft? B A. 7 ft 3 B. ft 3 C. ft 3 D. 3. One gallon fills about 3 in. 3. A rigt cylindrical carton is in. tall and olds 9 gal wen full. Find te radius of te carton to te nearest tent of an inc. G F. 0. G. 7. in. H in. J. 55. in. 3. Te eigt of a triangular prism is 8 feet. One side of te base measures 6 feet. Wat additional information do you need to find te volume? C A. te perimeter of te base B. te lengt of a second side of te base C. te altitude of te base to te 6-foot side D. te area of eac rectangular face of te prism. A rectangular prism as a volume of 00 ft 3. If te base measures 5 ft by 8 ft, wat is te eigt of te prism? F F..5 ft G..5 ft H. 0 ft J. 0 ft 5. How is te formula for finding te lateral area of a cylinder like te formula for finding te area of a rectangle? See margin. Mixed Review GO for Help Lesson -3 Lesson 0-6 Lesson 7-3 x Find te lateral area of eac figure to te nearest tent. 6. a rigt circular cone wit eigt mm and radius 5 mm 0. mm 7. a regular exagonal pyramid wit base edges 9. ft long and slant eigt 7 ft 69. ft Algebra Find te value of eac variable and te measure of eac labeled angle. 5; 60, 05, 95 50;,, 8, ; (c - 36) 0, 70 (6b + 0) (3c - 6) a (a - 30) (3c - ) (b + 5) (b - 5) (c + ) 5. You want to find te eigt of a tree near your scool. Your sadow is tree-fourts of your eigt. Te tree s sadow is 57 feet. How tall is te tree? Capter Surface Area and Volume 5. [] L.A. πr and A b; πr is te lengt of te base wen te cylinder is unwrapped. [] correct formulas are given, but comparison is unclear 630