Lesson 10.1 The Geometry of Solids

Transcription

1 Lesson 10.1 The Geometry of Solids For Eercises 1 1, refer to the figures below. O T A I D C J H Q P R B C E A B F G 1. The cylinder is (oblique, right). 2. OP is of the cylinder.. TR is of the cylinder.. Circles O and P are of the cylinder.. PQ is of the cylinder.. The cone is (oblique, right). 7. Name the base of the cone. 8. Name the verte of the cone. 9. Name the altitude of the cone. 10. Name a radius of the cone. 11. Name the type of prism. 12. Name the bases of the prism. 1. Name all lateral edges of the prism. 1. Name an altitude of the prism. In Eercises 1 17, tell whether each statement is true or false. If the statement is false, give a countereample or eplain why it is false. 1. The ais of a cylinder is perpendicular to the base. 1. A rectangular prism has four faces. 17. The bases of a trapezoidal prism are trapezoids. For Eercises 18 and 19, draw and label each solid. Use dashed lines to show the hidden edges. 18. A right triangular prism with height 19. An oblique trapezoidal pyramid equal to the hypotenuse Discovering Geometry Practice Your Skills CHAPTER 10

2 Lesson 10.2 Volume of Prisms and Cylinders In Eercises 1, find the volume of each prism or cylinder. All measurements are in centimeters. Round your answers to the nearest Right triangular prism 2. Right trapezoidal prism. Regular heagonal prism In Eercises, use algebra to epress the volume of each solid.. Right rectangular prism. Right cylinder;. Right rectangular prism base circumference p and half of a cylinder y y 2 h 2 7. You need to build a set of solid cement steps for the entrance to your new house. How many cubic feet of cement do you need? in. 8 in. ft CHAPTER 10 Discovering Geometry Practice Your Skills

3 Lesson 10. Volume of Pyramids and Cones In Eercises 1, find the volume of each solid. All measurements are in centimeters. Round your answers to two decimal places. 1. Rectangular pyramid; OP 2. Right heagonal pyramid. Half of a right cone P O 1 In Eercises, use algebra to epress the volume of each solid.... The solid generated by 0 b spinning ABC about 2a the ais A 7 2 C 2y B In Eercises 7 9, find the volume of each figure and tell which volume is larger. 7. A. B A. B A. B. 9 Discovering Geometry Practice Your Skills CHAPTER 10 7

4 Lesson 10. Volume Problems 1. A cone has volume 20 cm and height 1 cm. Find the radius of the base. Round your answer to the nearest 0.1 cm. 2. How many cubic inches are there in one cubic foot? Use your answer to help you with Eercises and.. Jerry is packing cylindrical cans with diameter in. and height 10 in. tightly into a bo that measures ft by 2 ft by 1 ft. All rows must contain the same number of cans. The cans can touch each other. He then fills all the empty space in the bo with packing foam. How many cans can Jerry pack in one bo? Find the volume of packing foam he uses. What percentage of the bo s volume is filled by the foam?. A king-size waterbed mattress measures 72 in. by 8 in. by 9 in. Water weighs 2. pounds per cubic foot. An empty mattress weighs pounds. How much does a full mattress weigh?. Square pyramid ABCDE, shown at right, is cut out of a cube with base ABCD and shared edge DE. AB 2 cm. Find the volume and surface area of the pyramid. E. In Dingwall the town engineers have contracted for a new water storage tank. The tank is cylindrical with a base 2 ft in diameter and a height of 0 ft. One cubic foot holds about 7. gallons of water. About how many gallons will the new storage tank hold? A D B C 7. The North County Sand and Gravel Company stockpiles sand to use on the icy roads in the northern rural counties of the state. Sand is brought in by tandem trailers that carry 12 m each. The engineers know that when the pile of sand, which is in the shape of a cone, is 17 m across and 9 m high they will have enough for a normal winter. How many truckloads are needed to build the pile? 8 CHAPTER 10 Discovering Geometry Practice Your Skills

5 Lesson 10. Displacement and Density 1. A stone is placed in a cm-diameter graduated cylinder, causing the water level in the cylinder to rise 2.7 cm. What is the volume of the stone? 2. A 11 g steel marble is submerged in a rectangular prism with base cm by cm. The water rises 0. cm. What is the density of the steel?. A solid wood toy boat with a mass of 2 g raises the water level of a 0 cm-by-0 cm aquarium 0. cm. What is the density of the wood?. For Awards Night at Baddeck High School, the math club is designing small solid silver pyramids. The base of the pyramids will be a 2 in.-by-2 in. square. The pyramids should not weigh more than pounds. One cubic foot of silver weighs pounds. What is the maimum height of the pyramids?. While he hikes in the Gold Country of northern California, Sid dreams about the adventurers that walked the same trails years ago. He suddenly kicks a small bright yellowish nugget. Could it be gold? Sid quickly makes a balance scale using his walking stick and finds that the nugget has the same mass as the uneaten half of his 0 g nutrition bar. He then drops the stone into his water bottle, which has a 2. cm radius, and notes that the water level goes up 0.9 cm. Has Sid struck gold? Eplain your reasoning. (Refer to the density chart in Lesson 10. in your book.) Discovering Geometry Practice Your Skills CHAPTER 10 9

6 Lesson 10. Volume of a Sphere In Eercises 1, find the volume of each solid. All measurements are in centimeters. Write your answers in eact form and rounded to the nearest 0.1 cm Cylinder with hemisphere taken out of the top A sphere has volume 221 cm. What is its diameter? 8. The area of the base of a hemisphere is 22 in 2. What is its volume? 9. Eight wooden spheres with radii in. are packed snugly into a square bo 12 in. on one side. The remaining space is filled with packing beads. What is the volume occupied by the packing beads? What percentage of the volume of the bo is filled with beads? 10. The radius of Earth is about 78 km, and the radius of Mercury is about 20 km. About how many times greater is the volume of Earth than that of Mercury? 70 CHAPTER 10 Discovering Geometry Practice Your Skills

7 Lesson 10.7 Surface Area of a Sphere In Eercises 1, find the volume and total surface area of each solid. All measurements are in centimeters. Round your answers to the nearest 0.1 cm If the surface area of a sphere is 8. cm 2, find its diameter.. If the volume of a sphere is cm, find its surface area. 7. Lobster fishers in Maine often use spherical buoys to mark their lobster traps. Every year the buoys must be repainted. An average buoy has a 12 in. diameter, and an average fisher has about 00 buoys. A quart of marine paint covers 17 ft 2. How many quarts of paint does an average fisher need each year? Discovering Geometry Practice Your Skills CHAPTER 10 71

8 LESSON 9. Circles and the Pythagorean Theorem 1. (2 2) cm 2, or about. cm 2 2. (72 2) cm 2, or about 9. cm 2. (8 7) cm.1 cm. Area.7 cm cm 2. AD 11.0 cm 10.7 cm. ST LESSON 10.1 The Geometry of Solids 1. oblique 2. the ais. the altitude. bases. a radius. right 7. Circle C 8. A 9. AC or AC 10. BC or BC 11. Right pentagonal prism 12. ABCDE and FGHIJ 1. AF, BG, CH, DI, EJ 1. Any of AF, BG, CH, DI, EJ or their lengths 1. False. The ais is not perpendicular to the base in an oblique cylinder. 1. False. A rectangular prism has si faces. Four are called lateral faces and two are called bases. 17. True LESSON 10. Volume of Pyramids and Cones cm cm. 1.7 cm. V 80. V 8 a2 b. V y 2 7. A: 128 cubic units, B: 1 cubic units. B is larger. 8. A: cubic units, B: cubic units. They have equal volumes. 9. A: 9 cubic units, B: 27 cubic units. B is larger. LESSON 10. Volume Problems 1.. cm in. 2 cans; 82 in 2.07 ft ;.% lb (about 1 ton). Note that AE AB and EC BC. V 8 cm ; SA (8 2) cm cm 2. About 110,7 gallons 7. 7 truckloads LESSON 10. Displacement and Density All answers are approimate cm g/cm. 0. g/cm..9 in.. No, it s not gold (or at least not pure gold). The mass of the nugget is 1 g, and the volume is 17.7 cm, so the density is 9. g/cm. Pure gold has density 19. g/cm. LESSON 10. Volume of a Sphere cm, or about 90.8 cm cm, or about. cm LESSON 10.2 Volume of Prisms and Cylinders cm 2. 1 cm. 1.9 cm. V y(2 ), or 8 2 y 12y. V 1 p2 h. V y 7. ft. 72 cm, or about 22.2 cm. 2 8 cm, or about 29. cm. 2 cm, or about 17.2 cm. 0 cm, or about 18. cm cm in 708. in in ; 7.% LESSON 10.7 Surface Area of a Sphere 1. V 1. cm ; S 1. cm 2 2. V 18. cm ; S 1. cm 2 Discovering Geometry Practice Your Skills ANSWERS 111

9 . V cm ; S 8.9 cm 2. V.1 cm ; S 1.1 cm 2. About.9 cm. About 7. cm quarts LESSON 11.1 Similar Polygons 1. AP 8 cm; EI 7 cm; SN 1 cm; YR 12 cm 2. SL.2 cm; MI 10 cm; md 120 ; mu 8 ; ma 80. Yes. All corresponding angles are congruent. Both figures are parallelograms, so opposite sides within each parallelogram are equal. The corresponding sides are proportional Yes. Corresponding angles are congruent by the CA Conjecture. Corresponding sides are proportional 2 = = No Yes. All angles are right angles, so corresponding angles are congruent. The corresponding side lengths have the ratio, 7 so corresponding side lengths are proportional A(0, 1) 8. to 1 y y B(2, ) D(2, ) C(1., 1.) D(2, 0.). CA cm 7. ABC EDC. Possible eplanation: A E and B D by AIA, so by the AA Similarity Conjecture, the triangles are similar. 8. PQR STR. Possible eplanation: P S and Q T because each pair is inscribed in the same arc, so by the AA Similarity Conjecture, the triangles are similar. 9. MLK NOK. Possible eplanation: MLK NOK by CA and K K because they are the same angle, so by the AA Similarity Conjecture, the two triangles are similar. LESSON 11. Indirect Measurement with Similar Triangles ft ft mi. About 18. ft. 0. m, 1.2 m, 1.8 m, 2. m, and.0 m LESSON 11. Corresponding Parts of Similar Triangles 1. h 0.9 cm; j.0 cm 2..7 cm,.0 cm,.0 cm. WX cm; AD 21 cm; DB 12 cm; YZ 8 cm; XZ 7.9 cm cm; y cm. a 8 cm; b.2 cm; c 2.8 cm. CB 2 cm; CD.2 cm; AD 8.7 cm LESSON 11. Proportions with Area 1.. cm cm :2 7. 2: cm tiles F(, 2) E(8, 2) LESSON 11. Proportions with Volume 1. Yes 2. No. 1 cm. 20 cm. 8:12. ft 2 LESSON 11.2 Similar Triangles 1. MC 10. cm 2. Q X; QR.8 cm; QS 11.2 cm. A E; CD 1. cm; AB 10 cm. TS 1 cm; QP 1 cm. AA Similarity Conjecture LESSON 11.7 Proportional Segments Between Parallel Lines cm 2. Yes. No. NE 1.2 cm. PR cm; PQ cm; RI 12 cm. a 9 cm; b 18 cm 112 ANSWERS Discovering Geometry Practice Your Skills