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Find the Volume of Each Pyramid. 1. SOLUTION 11-3 Volumes of Pyramids and Cones Find the volume of each pyramid. 1. 2. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs The base of this pyramid is a regular pentagon with of 9 inches and 5 inches and the height of the sides of 4.4 centimeters and an apothem of 3 pyramid is 10 inches. centimeters. The height of the pyramid is 12 centimeters. ANSWER: ANSWER: 75 in3 132 cm3 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. ANSWER: 62.4 m3 eSolutions Manual - Powered by Cognero Page 1 11-3 Volumes of Pyramids and Cones 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: 6. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. Use trigonometry to find the radius r. ANSWER: The volume of a circular cone is , or 298.7 m3 , where B is the area of the base, h is the Find the volume of each cone. Round to the height of the cone, and r is the radius of the base. nearest tenth. The height of the cone is 11.5 centimeters. 5. SOLUTION: ANSWER: The volume of a circular cone is , or 168.1 cm3 , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. ANSWER: 51.3 in3 eSolutions Manual - Powered by Cognero Page 2 11-3 Volumes of Pyramids and Cones 7. an oblique cone with a height of 10.5 millimeters and 8. a cone with a slant height of 25 meters and a radius a radius of 1.6 millimeters of 15 meters SOLUTION: SOLUTION: The volume of a circular cone is , or , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the Use the Pythagorean Theorem to find the height h of height is 10.5 millimeters. the cone. Then find its volume. ANSWER: So, the height of the cone is 20 meters. 3 28.1 mm ANSWER: 4712.4 m3 eSolutions Manual - Powered by Cognero Page 3 11-3 Volumes of Pyramids and Cones 9. HUTS The Caddo Indians, from the east part of STRUCTURE Find the volume of each pyramid. Texas, lived in tall cone-shaped grass huts made of Round to the nearest tenth. wooden pole frames with long prairie grasses dried and threaded through the poles in layers. These houses normally measured 40 feet tall. Suppose the diameter is also 40 feet. What is the volume inside the hut? 10. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. SOLUTION: The volume of a circular cone is , where B is the area of the base and h is the height of the cone. ANSWER: 605 in3 ANSWER: 11. about 16,755 ft3 SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 105.8 mm3 eSolutions Manual - Powered by Cognero Page 4 11-3 Volumes of Pyramids and Cones 12. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The apothem is . ANSWER: 482.1 m3 ANSWER: 233.8 cm3 13. 14. a pentagonal pyramid with a base area of 590 square SOLUTION: feet and an altitude of 7 feet The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base is a hexagon, so we need to make a right area of the base and h is the height of the pyramid. triangle to determine the apothem. The interior angles of the hexagon are 120°. The radius bisects the angle, so the right triangle is a 30°-60°-90° triangle. ANSWER: 1376.7 ft3 eSolutions Manual - Powered by Cognero Page 5 11-3 Volumes of Pyramids and Cones 15. a triangular pyramid with a height of 4.8 centimeters Use the Pythagorean Theorem to find the other leg a and a right triangle base with a leg 5 centimeters and of the right triangle and then find the area of the hypotenuse 10.2 centimeters triangle. SOLUTION: Find the height of the right triangle. The length of the other leg of the right triangle is 6 cm. So, the area of the base B is 24 cm2. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. ANSWER: 3 35.6 cm 16. A triangular pyramid with a right triangle base with a Therefore, the height of the triangular pyramid is 18 leg 8 centimeters and hypotenuse 10 centimeters has cm. a volume of 144 cubic centimeters. Find the height. ANSWER: SOLUTION: 18 cm The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. eSolutions Manual - Powered by Cognero Page 6 11-3 Volumes of Pyramids and Cones Find the volume of each cone. Round to the nearest tenth. 18. SOLUTION: 17. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the The volume of a circular cone is , cone. The radius of this cone is 4.2 centimeters and where r is the radius of the base and h is the height the height is 7.3 centimeters. of the cone. Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches. Therefore, the volume of the cone is about 134.8 cm3. ANSWER: 134.8 cm3 Therefore, the volume of the cone is about 235.6 in3. ANSWER: 235.6 in3 eSolutions Manual - Powered by Cognero Page 7 11-3 Volumes of Pyramids and Cones 20. 19. SOLUTION: SOLUTION: Use trigonometric ratios to find the height h and the radius r of the cone. Use a trigonometric ratio to find the height h of the cone. The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. cone. Therefore, the volume of the cone is about 1473.1 cm3. Therefore, the volume of the cone is about 2.8 ft3. ANSWER: ANSWER: 1473.1 cm3 2.8 ft3 eSolutions Manual - Powered by Cognero Page 8 11-3 Volumes of Pyramids and Cones 21. an oblique cone with a diameter of 16 inches and an 22. a right cone with a slant height of 5.6 centimeters and altitude of 16 inches a radius of 1 centimeter SOLUTION: SOLUTION: The cone has a radius r of 1 centimeter and a slant The volume of a circular cone is , where height of 5.6 centimeters. Use the Pythagorean r is the radius of the base and h is the height of the Theorem to find the height h of the cone. cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches. Therefore, the volume of the cone is about 1072.3 in 3. ANSWER: 1072.3 in3 Therefore, the volume of the cone is about 5.8 cm3. ANSWER: 5.8 cm3 eSolutions Manual - Powered by Cognero Page 9 11-3 Volumes of Pyramids and Cones 23. SNACKS Approximately how many cubic octagonal pyramid with a height of 5 feet. The base centimeters of roasted peanuts will completely fill a has side lengths of 2 feet. What is the volume of the paper cone that is 14 centimeters high and has a base greenhouse? diameter of 8 centimeters? Round to the nearest tenth. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, SOLUTION: the radius is or 4 centimeters. The height of the cone is 14 centimeters.
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