A cone is a solid formed by one circular base and a vertex.

Transcription

1 Volume of Cones Lesson 4.7 A cone is a solid fomed by one cicula base and a vetex. EXPLORE! Rita is making snow cones fo he fiends. She has fou cylindical glasses that she is using to take the cushed ice outside to he fiends that ae waiting to fill thei snow cone cups. Each snow cone cup is as tall as one of the glasses and has the same size cicula base. Rita wants to detemine if thee is a elationship between a cone and a cylinde. She needs to make sue she has enough ice fo 12 snow cones. Complete the Exploe! to find if Rita will have enough ice fo the snow cones. Step 1: Make o find a cone and a cylinde that have conguent bases and ae the same height. Step 2: Estimate how many times lage the volume of the cylinde is compaed to the volume of the cone. Step 3: Fill the cone with ice, beans o popcon kenels. Pou the contents of the cone into the cylinde. Repeat until the cylinde is full. Step 4: How many times did you need to empty the cone in ode to fill the cylinde? Step 5: What faction is the volume of the cone compaed to the volume of the cylinde? Step 6: What is the fomula fo the volume of a cylinde? Step 7: Combine the answes fom Step 5 and Step 6 to wite a fomula that can be used to find the volume of a cone. cones in a Cylinde Step 8: Use you fomula to find the volume of each cone. Use 3.14 fo π. Round to the neaest hundedth. a. 6 in b. 18 ft c. = 2.1 mm 15 in 7.5 ft 9.4 mm Step 9: Will Rita have enough cushed ice fo 12 snow cones if she fills the fou cylindes? 154 Lesson 4.7 ~ Volume of Cones

2 It takes the volume of thee cones to equal the volume of one cylinde when they have conguent bases and heights. The height, h, of a cone is how tall a cone is fom its vetex to the cente of its base. A pyamid and a pism with conguent bases and heights have the same elationship. h = h + h + h Example 1 Find the volume of the cone. Use 3.14 fo π. 4 cm 12 cm 12.6 cm Solution Wite the volume fomula fo a cone. V = 1_ 3 π²h Substitute all known values fo the vaiables. V 1_ 3 (3.14)(4)²(12) Find the value of the powe. V 1_ 3 (3.14)(16)(12) Multiply. V The volume of the cone is about cm³. Example 2 Solution Chantel helped with he siste s paty. Each child eceived a paty hat full of teats. Each hat had a volume of cubic inches. The adius of each hat was 3 inches. How tall was each paty hat? Wite the volume fomula fo a cone. V = 1_ 3 π²h Substitute all known values fo the vaiables _ 3 (3.14)(3)²h Find the value of the powe _ 3 (3.14)(9)(h) Multiply h Divide both sides of the equation by h h Each paty hat was about 7 inches tall. Lesson 4.7 ~ Volume of Cones 155

3 Example 3 Solution A cone-shaped popcon containe holds 314 cubic inches of popped con. The containe is 12 inches tall. Find the adius of the conical containe. Use 3.14 fo π. Wite the volume fomula fo a cone. V = 1_ 3 π²h Substitute all known values fo the vaiables _ 3 (3.14)²(12) Multiply ² Divide by on both sides of the equation ² Squae oot both sides of the equation. ± 25 ² 5 = The adius of the popcon containe is 5 inches. Execises Find the volume of each cone. Use 3.14 fo π. Round answes to the neaest whole numbe ft ft 8.2 cm 3. 3 cm 4 in 3 in 5 in 4. A cone has the same height as a cylinde. The aeas of the bases of the two solids ae the same. How many cones does it take to fill the cylinde? 5. The volume of a cylinde is 18 cubic metes. What is the volume of a cone with a conguent base and height? 6. A cylinde holds 87 gallons of wate. How many gallons of wate does a cone with the same adius and height hold? Explain you easoning. 7. Sketch a diagam of a cone with a adius of 9 yads and a height of 12 yads. Find the volume of the cone. Find the volume of each cone with the given adius and height. Use 3.14 fo π. Round to the neaest hundedth. 8. = 3 m 9. = 6 cm 10. d = 20 in h = 5 m h = 2 cm h = 7 in 156 Lesson 4.7 ~ Volume of Cones

4 Find the length of the adius in each cone. Use 3.14 fo π. Round to the neaest whole numbe. 11. V in³ 12. V ft³ 13. V in³ 9 in 4.5 ft 29.4 in 14. An oange cone is often used to alet dives in a constuction zone to wan them to slow down. A constuction cone is 24 inches tall. The cicula base has a diamete of 18 inches. Find the volume of the cone. Use 3.14 fo π. 15. An icicle is shaped like a cone. It has a diamete of 1 inch. It is 2 feet long. a. Daw a diagam and label it. b. What is the volume of the icicle in cubic inches? Use 3.14 fo π. Show all wok necessay to justify you answe. 16. A candle make makes cone-shaped candles. The candle make uses 9.42 cubic inches of wax to make each candle. The adius of each candle is 1.5 inches. Find the height of the candles. Use 3.14 fo π. 17. The conical pape cups next to a wate coole each hold 3.14 cubic inches of wate. The adius of each pape cup is 1 inch. Find the height of the cup. Use 3.14 fo π. 18. Fank is setting up a conical teepee. The box says it is 10 feet tall and has a volume of appoximately cubic feet. He needs to find the diamete of the teepee to find a space lage enough to set it up. a. What is the diamete of the teepee? b. If Fank has a squae patch of gass in his yad that has a peimete of 28 feet, will the teepee fit in it? Explain you easoning. Find the volume of each composite solid. Use 3.14 fo π. Round to the neaest hundedth ft 2.5 ft ft 8 ft 12 ft 14 ft Lesson 4.7 ~ Volume of Cones 157

5 Review Find the distance between each pai of points. If necessay, ound to the neaest tenth. 21. (2, 7) and (5, 11) 22. (6, 0) and ( 2, 15) 23. ( 3, 2) and ( 10, 5) Find the volume of each cylinde. Use 3.14 fo π ft cm 8 cm Simplify. 16xy⁸ 27. 5m³ ( 2a⁵ ) k- ⁵w⁷ k⁹w - ³ x³y¹⁰ Tic-Tac-Toe ~ Te ach i ng Volume Design an activity kit to teach thid gades about volume. The activity kit could do any combination of the following: Teach the geneal concept of volume Show volume using diffeent mateials Show and teach that diffeent sizes of containes have diffeent volumes Show and teach that sometimes containes that ae diffeent shapes have the same volume The kit will include: A witten explanation descibing what students will lean fom the lesson. A step-by-step lesson plan to follow that will take appoximately 30 minutes to complete. Mateials to use fo the activity. Packaging all pats of the kit ae togethe in some sot of package. If possible, teach you lesson on volume to a goup of thid gades. 158 Lesson 4.7 ~ Volume of Cones

6 Tic-Tac-Toe ~ Displ ace m e n t A wate glass is half full. When ice cubes ae added the wate level ises. This is called displacement. It is possible to calculate the volume of objects using displacement. Example: A cylinde has a adius of 1.5 inches. The 8-inch tall cylinde is half full of wate. When ice is added the wate level ises 3 inches. Find the volume of the ice. V = π²h Substitute known values. = (π)(1.5)²(3) Find the value of the powe. (3.14)(2.25)(3) Multiply The ice has a volume of about cubic inches. Find the volume of each object using displacement. Use 3.14 fo π. 1. A 12-inch tall cylinde has a adius of 3 inches. A ock is dopped in the cylinde and the wate ises 5 inches. Find the volume of the ock. 2. Keely filled he glass half full of tea. She then put in 5 equal-sized ice cubes. The base of he glass has a 10 cm diamete. The tea eached 16 cm high in the glass. The tea ose to 20 cm when the ice was added. a. What was the height change when the ice was added? b. Find the volume of the ice. c. Find the volume of one ice cube. 3. A fish tank is a ectangula pism. The wate ises 5 inches when the fish ae added. Find the volume of the fish. (Note: 16 is the height of the fish tank.) 4. Do you own displacement tests. Find a cylinde o pism to use. a. Calculate the aea of the base. Name of b. Patially fill with wate. Height Change Object c. Measue the height of the wate. d. Find fou objects and ecod them in the table. e. Find the height change of the wate when each object is dopped in. f. Calculate the volume of each object. Show all wok in you table. Wok Volume of Object Lesson 4.7 ~ Volume of Cones 159