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Find the volume of each.

Find the volume of each. V = ( Bh )/3 = π(4 2 )(8)/3 = 42.7π mm 3. V = ( Bh )/3 = (5×9)(12)/3 = 180 ft 3. Find the height of a hexagonal pyramid with a base area of 130 square meters and a volume of 650 cubic meters.

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Find the volume of each.

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  1. Find the volume of each. V = (Bh)/3 = π(42)(8)/3 = 42.7π mm3 V = (Bh)/3 = (5×9)(12)/3 = 180 ft3 Find the height of a hexagonal pyramid with a base area of 130 square meters and a volume of 650 cubic meters. Find the volume of a cone with a diameter of 8.4 meters and a height of 14.6 meters. V = (Bh)/3 650 = (130)h/3 = 15 m V = (Bh)/3 = π(√(4.22)(14.6)/3 = 85.8π m3

  2. Learning Target: I will be able to solve problems involving the volume and surface areas of spheres. Standard 9.0 Students compute the volume and surface area of spheres. Ch 12.6Volumes & Surface Areas of Spheres

  3. great circle great circle A circle formed when a plane intersects with its center at the center of the sphere. pole The endpoints of the diameter of a great circle. hemisphere One of two congruent parts into which a great circle separates a sphere. Vocabulary

  4. Theorem 12-13 Concept

  5. Surface Area of a Sphere Find the surface area of the sphere in terms of π. S = 4r2 Surface area of a sphere = 4(4.5)2 r = 4.5. ≈ 81π Multiply. Answer: about 254 .6 in2 Example 1

  6. Find the surface area of the sphere. A. 147.2π in2 B. 150.5π in2 C. 153.6π in2 D. 156.3π in2 S = 4r2 Surface area of a sphere = 4(6.25)2 r = 4.5. = 156.3π Multiply. Example 1

  7. Use Great Circles to Find Surface Area A. Find the surface area of the hemisphere. Find half the area of a sphere with the radius of 3.7 millimeters. Then add the area of the great circle. Surface area of a hemisphere r = 3.7. = 41π Multiply. Answer: about 129.0 mm2 Example 2A

  8. Use Great Circles to Find Surface Area B. Find the surface area of a sphere if the circumference of the great circle is 10 feet. First, find the radius. The circumference of a great circle is 2r. So, 2r = 10 or r = 5. S = 4r2 Surface area of a sphere = 4(5)2 r = 5. = 100π Multiply. Answer: about 314.2 ft2 Example 2B

  9. Use Great Circles to Find Surface Area C. Find the surface area of a sphere if the area of the great circle is approximately 220 m2. First, find the radius. The area of a great circle is r2. So, r2 = 220 or r=√220/π. S = 4r2 Surface area of a sphere = 4(√220/π)2 r = √(220/π) = 880 Multiply. Answer: 880 m2 Example 2C

  10. A. Find the surface area of the hemisphere. A. 35.3π m2 B. 52.9π m2 C. 53.9π m2 D. 55.0π m2 Surface area of a hemisphere (4.2)2 (4.2)2 r = 4.2 = 52.92π Multiply. Example 2A

  11. B. Find the surface area of a sphere if the circumference of the great circle is 8 feet. A. 32π ft2 B. 64π ft2 C. 128π ft2 D. 256π ft2 S = 4r2 Surface area of a sphere = 4(4)2 r = 8π/2π. = 64π Multiply. Example 2B

  12. C. Find the surface area of the sphere if the area of the great circle is approximately 160 square meters. A. 320 ft2 B. 440 ft2 C. 640 ft2 D. 720 ft2 S = 4r2 Surface area of a sphere = 4(√160/π)2 r = √(160/π) = 640 ft2 Multiply. Example 2C

  13. Theorem 12-14 Concept

  14. (15)3r = 15 Volumes of Spheres and Hemispheres A. Find the volume a sphere with a great circle circumference of 30 centimeters in terms of π. Find the radius of the sphere. The circumference of a great circle is 2r. So, 2r = 30 or r = 15. Volume of a sphere = 4500π cm3 Multiply. Answer: about 14,137.2 cm3. Example 3A

  15. r 3 Volumes of Spheres and Hemispheres B. Find the volume of the hemisphere with a diameter of 6 feet. Round to the nearest tenth. The volume of a hemisphere is one-half the volume of the sphere. Volume of a hemisphere = 18π ft3 Multiply. Answer: about 56.5 cubic feet. Example 3B

  16. (8)3r = 8 A. Find the volume of the sphere. A. 85.3π cm3 B. 511.8π cm3 C. 682.7π cm3 D. 2047.2π cm3 Volume of a sphere = 682.7π cm3 Multiply. Example 3A

  17. r 3 B. Find the volume of the hemisphere. A. 1066π m3 B. 2132π m3 C. 8554π m3 D. 42667π m3 Volume of a hemisphere (40)3 = 42667π m3 Multiply. Example 3B

  18. Solve Problems Involving Solids ARCHEOLOGYThe stone spheres of Costa Rica were made by forming granodiorite boulders into spheres. One of the stone spheres has a volume of about 36,000 cubic inches. What is the diameter of the stone sphere? Understand You know that the volume of the stone is 36,000 cubic inches. Plan First use the volume formula to find the radius. Then find the diameter. Example 4

  19. Solve Volume of a sphere Divide each side by Use a calculator to find Solve Problems Involving Solids Replace V with 36,000. 2700 ( 1 ÷ 3 ) ENTER 30 The radius of the stone is 30 inches. So, the diameter is 2(30) or 60 inches. Answer: 60 inches Example 4

  20. If the diameter is 60, then r = 30. If r = 30, then V = cubic inches. The solution is correct. Solve Problems Involving Solids CHECK You can work backward to check the solution.  Example 4

  21. 2 3 d 2 4000π = π3 RECESS The jungle gym outside of Jada’s school is a perfect hemisphere. It has a volume of 4,000 cubic feet. What is the diameter of the jungle gym? A. 10.7 feet B. 12.6 feet C. 14.4 feet D. 36.3 feet Volume of a hemisphere V = 4000π, r = d/2 = 36.3 ft3 Multiply. Example 4

  22. Concept

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