11.7 Suface Aeas and Volumes of Cones Essential Question How can you find te suface aea and te volume of a cone? Finding te Suface Aea of a Cone Wok wit a patne. Constuct a cicle wit a adius of 3 inces. Mak te cicumfeence of te cicle into six equal pats, and label te lengt of eac pat. Ten cut out one secto of te cicle and make a cone. 3 in. a. Explain wy te base of te cone is a cicle. Wat ae te cicumfeence and adius of te base? b. Wat is te aea of te oiginal cicle? Wat is te aea wit one secto missing? c. Descibe te suface aea of te cone, including te base. Use you desciption to find te suface aea. Finding te Volume of a Cone Wok wit a patne. Te cone and te cylinde ave te same eigt and te same cicula base. Wen te cone is filled wit sand and poued into te cylinde, it takes tee cones to fill te cylinde. CONSTRUCTING VIABLE ARGUMENTS To be poficient in mat, you need to undestand and use stated assumptions, definitions, and peviously establised esults in constucting aguments. Use tis infomation to wite a fomula fo te volume V of a cone. Communicate You Answe 3. How can you find te suface aea and te volume of a cone? 4. In Exploation 1, cut anote secto fom te cicle and make a cone. Find te adius of te base and te suface aea of te cone. Repeat tis tee times, ecoding you esults in a table. Descibe te patten. Section 11.7 Suface Aeas and Volumes of Cones 641
11.7 Lesson Wat You Will Lean Coe Vocabulay lateal suface of a cone, p. 642 Pevious cone net composite solid 2 slant eigt Find suface aeas of igt cones. Find volumes of cones. Use volumes of cones. Finding Suface Aeas of Rigt Cones Recall tat a cicula cone, o cone, as a cicula base and a vetex tat is not in te same plane as te base. Te altitude, o eigt, is te pependicula distance between te vetex and te base. In a igt cone, te eigt meets te base at its cente and te slant eigt is te distance between te vetex and a point on te base edge. Te lateal suface of a cone consists of all segments tat connect te vetex wit points on te base edge. Wen you cut along te slant eigt and lay te igt cone flat, you get te net sown at te left. In te net, te cicula base as an aea of 2 and te lateal suface is a secto of a cicle. You can find te aea of tis secto by using a popotion, as sown below. Aea of secto Aea of cicle = Aea of secto 2 Ac lengt Cicumfeence of cicle = 2 2 Set up popotion. Substitute. Aea of secto = 2 2 2 Multiply eac side by 2. Aea of secto = Simplify. Te suface aea of a igt cone is te sum of te base aea and te lateal aea,. Coe Concept Suface Aea of a Rigt Cone Te suface aea S of a igt cone is S = 2 + wee is te adius of te base and is te slant eigt. slant eigt base vetex eigt lateal suface Find te suface aea of te igt cone. Finding Suface Aeas of Rigt Cones 6 in. 4 in. S = 2 + = 4 2 + (4)(6) = 40 125.66 Te suface aea is 40, o about 125.66 squae inces. Monitoing Pogess 1. Find te suface aea of te igt cone. Help in Englis and Spanis at BigIdeasMat.com 10 m 7.8 m 642 Capte 11 Cicumfeence, Aea, and Volume
Finding Volumes of Cones Conside a cone wit a egula polygon inscibed in te base. Te pyamid wit te same vetex as te cone as volume V = 1 B. As you incease te numbe of sides of 3 te polygon, it appoaces te base of te cone and te pyamid appoaces te cone. Te volume appoaces 1 3 2 as te base aea B appoaces 2. Coe Concept Volume of a Cone Te volume V of a cone is V = 1 3 B = 1 3 2 wee B is te aea of te base, is te eigt, and is te adius of te base. B B Find te volume of te cone. Finding te Volume of a Cone 4.5 cm V = 1 3 2 Fomula fo volume of a cone = 1 3 (2.2)2 4.5 Substitute. = 7.26 Simplify. 22.81 Use a calculato. Te volume is 7.26, o about 22.81 cubic centimetes. 2.2 cm Monitoing Pogess Find te volume of te cone. 2. Help in Englis and Spanis at BigIdeasMat.com 3. 5 m 13 in. 8 m 7 in. Section 11.7 Suface Aeas and Volumes of Cones 643
Using Volumes of Cones Finding te Volume of a Simila Solid Cone A and cone B ae simila. Find te volume of cone B. Cone A Cone B Cone C 8 cm V = 15 ft 3 Radius of cone B Te scale facto is k = Radius of cone A = 9 3 = 3. Use te scale facto to find te volume of cone B. 3 ft Volume of cone B Volume of cone A = k3 Te atio of te volumes is k 3. Volume of cone B = 3 15 Substitute. Volume of cone B = 405 Solve fo volume of cone B. 9 ft V = 384 cm 3 Cone D 2 cm Te volume of cone B is 405 cubic feet. Monitoing Pogess Help in Englis and Spanis at BigIdeasMat.com 4. Cone C and cone D ae simila. Find te volume of cone D. Find te volume of te composite solid. Finding te Volume of a Composite Solid 4 m Let 1 be te eigt of te cylinde and let 2 be te eigt of te cone. 6 m 5 m Volume of solid = Volume of cylinde + Volume of cone 5 cm = 2 1 + 1 3 2 2 Wite fomulas. = 6 2 5 + 1 3 62 4 Substitute. = 180 + 48 Simplify. = 228 Add. 716.28 Use a calculato. Te volume is 228, o about 716.28 cubic metes. 10 cm Monitoing Pogess 5. Find te volume of te composite solid. Help in Englis and Spanis at BigIdeasMat.com 644 Capte 11 Cicumfeence, Aea, and Volume
11.7 Execises Dynamic Solutions available at BigIdeasMat.com Vocabulay and Coe Concept Ceck 1. WRITING Descibe te diffeences between pyamids and cones. Descibe tei similaities. 2. COMPLETE THE SENTENCE Te volume of a cone wit adius and eigt is 1 te volume of 3 a(n) wit adius and eigt. Monitoing Pogess and Modeling wit Matematics In Execises 3 6, find te suface aea of te igt cone. (See Example 1.) 3. 8 in. 16 in. 4. 11 cm 7.2 cm 5. A igt cone as a adius of 9 inces and a eigt of 12 inces. 6. A igt cone as a diamete of 11.2 feet and a eigt of 9.2 feet. In Execises 7 10, find te volume of te cone. (See Example 2.) 7. 13 mm 10 mm 8. 1 m 2 m In Execises 13 and 14, te cones ae simila. Find te volume of cone B. (See Example 3.) 13. 14. Cone A 8 ft V = 32 ft 3 Cone A V = 120 m 3 10 m Cone B 16 ft Cone B 4 m In Execises 15 and 16, find te volume of te composite solid. (See Example 4.) 9. A cone as a diamete of 11.5 inces and a eigt of 15.2 inces. 15. 10 cm 16. 5.1 m 10. A igt cone as a adius of 3 feet and a slant eigt of 6 feet. 5.1 m 5.1 m In Execises 11 and 12, find te missing dimension(s). 11. Suface aea = 75.4 cm 2 12. Volume = 216 in. 3 18 in. 17. ANALYZING RELATIONSHIPS A cone as eigt and a base wit adius. You want to cange te cone so its volume is doubled. Wat is te new eigt if you cange only te eigt? Wat is te new adius if you cange only te adius? Explain. Section 11.7 Suface Aeas and Volumes of Cones 645
18. HOW DO YOU SEE IT A snack stand seves a small ode of popcon in a cone-saped containe and a lage ode of popcon in a cylindical containe. Do not pefom any calculations. 3 in. 3 in. 8 in. 8 in. $1.25 $2.50 a. How many small containes of popcon do you ave to buy to equal te amount of popcon in a lage containe? Explain. b. Wic containe gives you moe popcon fo you money? Explain. In Execises 19 and 20, find te volume of te igt cone. 19. 22 ft 60 20. 14 yd 32 23. REASONING To make a pape dinking cup, stat wit a cicula piece of pape tat as a 3-inc adius, ten follow te given steps. How does te suface aea of te cup compae to te oiginal pape cicle? Find m ABC. 3 in. fold fold B open cup 24. THOUGHT PROVOKING A fustum of a cone is te pat of te cone tat lies between te base and a plane paallel to te base, as sown. Wite a fomula fo te volume of te fustum of a cone in tems of a, b, and. (Hint: Conside te missing top of te cone and use simila tiangles.) a b A C 21. MODELING WITH MATHEMATICS A cat eats alf a cup of food, twice pe day. Will te automatic pet feede old enoug food fo 10 days? Explain you easoning. (1 cup 14.4 in. 3 ) 2.5 in. 7.5 in. 25. MAKING AN ARGUMENT In te figue, te two cylindes ae conguent. Te combined eigt of te two smalle cones equals te eigt of te lage cone. You fiend claims tat tis means te total volume of te two smalle cones is equal to te volume of te lage cone. Is you fiend coect? Justify you answe. 4 in. 22. MODELING WITH MATHEMATICS Duing a cemisty lab, you use a funnel to pou a solvent into a flask. Te adius of te funnel is 5 centimetes and its eigt is 10 centimetes. You pou te solvent into te funnel at a ate of 80 millilites pe second and te solvent flows out of te funnel at a ate of 65 millilites pe second. How long will it be befoe te funnel oveflows? (1 ml = 1 cm 3 ) Maintaining Matematical Poficiency Find te indicated measue. (Section 11.2) 26. CRITICAL THINKING Wen te given tiangle is otated aound eac of its sides, solids of evolution ae fomed. Descibe te tee solids and find tei volumes. Give you answes in tems of. 27. aea of a cicle wit a adius of 7 feet 28. aea of a cicle wit a diamete of 22 centimetes 29. diamete of a cicle wit an aea of 30. adius of a cicle wit an aea of 256 squae metes 529 squae inces 15 25 20 Reviewing wat you leaned in pevious gades and lessons 646 Capte 11 Cicumfeence, Aea, and Volume