13. a. right triangular prism

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1 C H A P T E R 0 SURFACE AREAS AND VOLUMES LESSON 0- pp Surface area measures the boundary of a -dimensional figure, while volume measures the interior.. volume. surface area. volume 5. surface area 6. a. Sample reduced size: b. Surface area cm 7. a. Lateral area is the total area of the lateral surface of a solid. b. L.A. ph 8. The lateral area of a right cylinder is the product of its height and the circumference of the base. 9. In a prism or cylinder, S.A. L.A. B 0. L.A. ph cm S.A. L.A. B cm. hypotenuse of base L.A. ph in. S.A. L.A. B in.. a. cube b. L.A. area of one face units c. S.A. 6area of one face units 0 0. a. right triangular prism b. L.A cm c. S.A cm. Surface area cm 5. Surface area LW LH WH 6. Surface area 6s 7. Lateral area ph cm 8. a. Sample: b. No. The formula does not hold for an oblique cylinder because the net for an oblique cylinder is not a quadrilateral and does not have constant height. 9. a. Surface area units b. Using dimensions 0,,, surface area units c. 856 d. Using dimensions, 9,, surface area units Using dimensions, 8,, surface area units Divide: L.A. ph m S.A. L.A. B ,7.5 m gallons needed for one coat of paint:, gallons needed for two coats of paint: About 6 gallons of paint are needed. 6

2 . Sample:. Given: Regular hexagon ABCDEF To prove: AEF BDC Argument: Conclusions Justifications 0. Regular hexagon Given 0. ABCDEF. AF BC; EF DC; definition of 0. F C regular polygon. AEF BDC SAS Congruence Theorem. To locate the center of a regular nonagon, find the point of intersection of either the perpendicular bisectors of any two sides or the bisectors of any two angles Answers will vary. Sample answers for one paper bag are given. a. 7 in. wide by in. long by 7 in. high b. S.A. L.A. B ph B square inches c. 7 " 7 " " " o v e rla p a re a d. Sum of areas of paper used in base of bag in. Area of base in part b in. Difference: in. 6. Sample: Start with a square base, 5 in. by 5 in. Areabase in., so you have in. left for the lateral area. L.A. ph h 0h, so 75 0h, and h in. The dimensions 0 are 5 in. wide by 5 in. long by.75 in. high. IN-CLASS ACTIVITY p Area ABC Area ACD Area ADE Area ABE units. a. Area sl sl sl sl sl sl sl b. p 6s; lp l6s sl; yes LESSON 0- pp units. a. sl b. yes. True. L.A. lateral area l slant height p perimeter of base 5. L.A. lateral area l length of any edge of the cone p circumference of base 6. L.A. lp cm 7. a. m 6 m 6

3 7. b. slant height 08, 0,9,66 79 meters c. p 6 86 m L.A. lp ,00 square meters 8. A right cone is the limit of regular pyramids as the number of sides of its base increases without bound yd 0.9 m 09.7 m yd m yd 8.8 m yd Areafootball field square meters Lateral area of pyramid of Khufu 86,00 square meters Divide areas: 86, The lateral area of the pyramid of Khufu is about 6 times as large as the area of a football field. 0. a. slant height 9 0. units b. L.A. lp units c. S.A. L.A. B units. a lateral area. a. slant height b. The slant height is the length of the hypotenuse of a right triangle, while the height is the length of a leg of the right triangle.. a. Sample: 50 b. L.A. pl units c. S.A. L.A. B units. a. Sample: 6 6 b. L.A. pl units c. S.A. L.A. B units 5. a. Sample: b. Let l length of any edge of the cone. l L.A. pl units c. S.A. L.A. B units 6. a. Areashaded region AreaB 8 76 units b. Lateral area Areashaded region 76 units c. length of ATC CircumferenceB units

4 6. d. Circumferencebase r 7 r r 7.5 Areabase r units 7. Using dimensions, 5, and 8, surface area square inches, or square feet 9.5 square feet 8. a. Areametal Area of circle with radius.75 cm cm b. Areacardboard cm 9. Sample: 0. length of lateral edge cm. Uniqueness Property: Every polygonal region has a unique area. Rectangle Formula: The area of a rectangle with dimensions l and w is lw. Congruence Property: Congruent figures have the same area. Additive Property: If A and B are nonoverlapping regions, then AreaA B AreaA AreaB.. a. 6 b. slope of t c. The slope of a line perpendicular to t is, since.. a. x 6 x 6 b. y 6 y 6 c. z 6 z y a. The cone with the 0 sector removed is the tallest. b. The cone with the 5 sector removed has the greatest lateral area. c. The cone with the 5 sector removed has a base with the greatest area. d. As the measure of the central angle increases and a cone is formed, the height of the cone increases, its lateral area decreases, and its base area decreases. LESSON 0- pp Volume ,00 cm. No. Yes. Volume in. 6 oz 5. pt pt gal oz t 8, x 6, - in in. gal 6

5 6. A two-liter container contains 000 milliliters. 7. x is a cube root of y if and only if x y a. s 0 s 0 cm b. 0. cm. To check, cube your answer to see if you get the number you started with.. a. Areatop 7 8 in. The top adds 8 in. to the bag s surface area. b. No change in the volume. 5. V lwh 576 8h h Its height is inches a. B area of a base b. The area of the base is lw, so the formula V Bh is equivalent to V lwh. 8. Let s length of an edge. s 9,79 cm Areaone face s 96 cm Surface area cm 9. a. Surface area of smaller cube 6x Surface area of larger cube 6x 66x 96x Ratio: 6x 96x 6 b. Volume of smaller cube x Volume of larger cube x 6x x Ratio: 6x 6 0. a. Volume cm b. cubic inch 6.9 cubic centimeters. a. yard feet b. square yard 9 square feet c. cubic yard 7 cubic feet. S.A. L.A. B pl s sl s sl s. a. S.A. L.A. B in. b. 90. You would need to buy square feet of adhesive paper.. L.A. pl units 5. a. HJ JG HG HJ 5 6 HJ 5 6 HJ HJ HI HJ Area GHI ft b. S.A. sides bases floor ft 6. Samples: toothpaste tubes, perfume bottles 7. a. Samples: a b c b. greatest surface area: a and c c. least surface area: b 65

6 LESSON 0- pp Sample: a. x x b. x x x x x x x. a. x 8x b. x 8x 6x 6x x 88 6x 9x 88. a. r s tx y z b. r s tx y z rx ry rz sx sy sz tx ty tz 5. a. u vw xy z b. uwy uwz uxy uxz vwy vwz vxy vxz c. u vw x uw ux vw vx; y zuw ux vw vx uwy uwz uxy uxz vwy vwz vxy vxz Yes, the product checks. 6. e e a a b a b 7.8 c c c e f f c d 7. x 9x x 8x x 7 x x 7 b a b d e d f d f 8. a 57a 6 a 5a 8a 0 a 5a 0 9. a a a a a a a a a 9a a 6 a 6a a 6 0. a b a 8b a 8ab ab 8b 6a 6b a 6a ab 6b 8b. The volume is multiplied by 5.. The volume is multiplied by 6.. Let l, w, and h be the original dimensions, with V lwh. When 6 is added to the length, then V l 6wh lwh 6wh. So the volume is increased by six times the product of the height and the width.. a. If volume of Y lwh, then volume of X.h.w.l =.06lwh. Bag X holds about twice as much as bag Y. b. Bag X should cost about twice as much as bag Y. 5. width x, since x 5x x 8x 5 6. a. Sample: b a b a a b Volume a b a a b ab b b. a ba ba b a ab b a b a a b ab a b ab b a a b ab b 7. Let l, w, and h be the original dimensions. h wl.lwh h.h.h h cm The original height was cm. 8. c Double the thickness, halve the height. 66

7 9. Yes. If s is the side length of the cube and s 0, then ss 5s s. 0. a. gallon cubic inches b. foot inches, so cubic foot 78 cubic inches. 78 cubic inches c. cubic foot cubic foot gallon 78 cubic inches gallons 7.8 gallons cubic foot about 7.8 gallons.6. If the volume is 7 in., then the length of one side is inches. The surface area is 6 5 in.. height of single card cm Volume cm. S.A. L.A. B pl B units 5. Let x be the unknown leg length. x x 0.6 x 0.6 x 0.8 Lateral area units 6. r 0 r 0 0 r.8 7. a. sides top bottom inside square units b. 6 cubes c. 8 cubes d. 0 cubes e. 0 cubes 8. a. outside inside square units b. 8 cubes c. 0 cubes d. 0 cubes e. 0 cubes LESSON 0-5 pp A cubic foot of liquid is about 7.8 gallons.. V Bh ,000 ft 6,000 ft 7.8 gal 5,076,000 gal ft The cylindrical tank can hold about 5,076,000 gallons of oil.. d. V Bh m 5. V Bh ft 6. h 8 h 6 8 h 0 h 0 0 V Bh units 7. V Bh units 8. V Bh 7 units 9. V Bh 70 0 feet 0. Cavalieri s Principle was discovered by Italian and Chinese mathematicians.. Cavalieri s Principle: Let I and II be two solids included between parallel planes. If every plane P parallel to the given planes intersects I and II in sections with the same area, then VolumeI VolumeII.. V Bh 5 75,8 m Figure out how many liters are in m, given that L 000 cm : L 00 cm 00 cm 00 cm 000 L 000 cm m m m m,8 m 000 L m The tank holds about liters of oil. 67

8 . V Bh r h. Their heights are the same. Both are 8 times the height of a penny. 5. V Bh 8 B B The area of each base is 9.5 m. 6. V Bh 9 89 m 7. a. Yes b. For the second glass, r..65, and V Bh.65 h.75h For the first glass, r..5, and V Bh.5 h.5h Since , the second glass can hold more than twice as much as the first glass. 8. The plane sections parallel to the base do not have the same area. 9. First cylinder: V r h Second cylinder: V 0.5r h 0.5r h Divide volumes: First cylinder Second cylinder r h 0.5r h 0.5 The first cylinder has four times the volume of the second cylinder. 0. V Bh 5 5 cm grams. x 5yx y 7 x 5xy 8xy 0y 8x 5y x 8x xy 5y 0y. The volume is multiplied by cm 0 cm of lead was used to make the box.. S.A. L.A. B ph B 5 56 square units 5. A hb b, trapezoid; A bh, parallelogram; A lw, rectangle; A s, square 6. Answers may vary. In theory, LR can be made as long as one wishes, though in practice it is difficult to make LR much longer than the width of two pennies. LESSON 0-6 pp The best formulas to remember are the formulas that apply to the most figures.. boxes, cylinders, prisms. cones, pyramids. a. L.A. ph L.A. lp S.A. L.A. B S.A. L.A. B V Bh b. S.A. r V Bh V r 5. L.A. lateral area of the prism or cylinder p perimeter of its base h its height 6. Substitute the circle formulas B r and p r in the corresponding formulas for pyramids and prisms. 7. Many mathematicians recall formulas by trying to prove them from simpler formulas known to be true. 8. L.A. ph rh 9. V Bh r h 0. S.A. L.A. B rh r. a. L.A. lp lr rl b. S.A. L.A. B rl r rl r. a. S.A. 6t 6t t units b. V t 8t units. a. S.A. L.A. B ph B 6 0x 6 8x 9 9x 9 units b. S.A. L.A. B 68

9 . a. L.A. lp 7q y 7qy units b. L.A. lp or L.A. rl 5. a. V Bh 6 z8w wz units b. V Bh 6. a. V Bh units b. x Find length x in the diagram: x 5 5 L.A. ph units c. S.A. L.A. B units 7. V r h 9,66 7 h 9,66 79h 5 h The cylinder s height is 5 cm. 8. x y x yx y x xy xy y x xy y 9. original cube: V 8 5 units resulting box: V 88 k8 k 86 k 5 8k The volume is decreased by 8k. x 0. a. slant height of sides: slant height of other sides: b. L.A ,500 m L.A. of Pyramid of the Sun 5,500 m L.A. of pyramid of Khufu 86,00 m 5 or 6% 8 The Mexico pyramid has lateral area about 5 8 that of the Khufu pyramid.. Area of shaded ring Area of large circle Area of small circle R r R r. a. mc b. BC BA 7, since ma mc c. AC slope of line l b 0 0 a b a. y 0, y = - x + 6, 0 x k 8 k 69

10 5. a., b. Samples: If r 8 and h.5, then S.A. L.A. B pl B S.A S.A units If r 5 and h 5, then S.A. L.A. B pl B S.A S.A units IN-CLASS ACTIVITY p. 598.,. Check students work.. Sample: D F A G D-AGC from n et I. Sample: C F E A 8 G E D C.5 LESSON 0-7 pp Sample: They are formed from the same net.. They are the same pyramid.. They have congruent bases and equal heights. D G G-DEF or D-GEF from n et II F A 5 5 D G D-AGF from n et III. The volume of the pyramid is the volume of the box. 5. V Bh 8.5 ft 6. V Bh units 7. V Bh units 8. V Bh units 9. V Bh 0 r 5 r r.8 cm 0. A cone and a cylinder have identical bases and equal heights. If the volume of the cylinder is V, then the volume of the cone is V.. V Bh r h r h. 5 acres,560 ft,960,00 ft acre V Bh,960,0077 6,000,000 ft. The volume is multiplied by.8.. The volume is multiplied by 7, or a. V Bh 8 75 cm b. L 000 cm 000 cm 75 cm. The cup will need to be used times in order to fill a liter jug. c. Let l slant height of the cone. l 8 7 L.A. pl cm 70

11 6. V Bh s 0 s The length of a base edge is 0 cm. 7. S.A At least 87 square inches of paper are needed. 8. L.A. 9. V Bh l s 6 9 6ls 8 6 units AC S.A. L.A. B ph B units 0. a. V x y 8z 5 b. V 8zx z xyz yz 0x 5xy 5y 0. Volume of wood lost in the planing k k 8k k k k k 5 6 k 0.6 6% 8k About 6% is lost in the planing.. a. Answers may vary. b. Answers may vary. c. three times LESSON 0-8 pp Prism-Cylinder Volume Formula; Pyramid-Cone Volume Formula; Sphere Volume Formula. Archimedes discovered the Sphere Volume Formula.. a. radius Areashaded cross section 0 0 units b. Areashaded ring units. a. Volumecylinder Bh units b. Volumetwo cones Bh units c. Volumesolid region units d. Volumesphere r units 5. a. Sample reduced size: b. V r 6 cm 6. d between 00 and 00 cubic inches 7. V r 68 r 0 r 0 r 6 The radius is about meters. 8. cube: V s in. sphere: V r in % 80.6 The ball fills about 5% of the box. cm 7

12 9. a. hemisphere: V r 8 cm cone: V Bh 0 0 cm ice cream: V cm b % 8 You have eaten 7.5% of the ice cream. 0. a. Volumetank 8.7 cubic meters The town uses about 500 m and replaces 00 m each day, so the tank loses m of water each day The full tank would last 0.7, or about, days. b The full tank would last., or about, days.. V r d d 8 6 d units. one cone: V Bh units solid two cones: V units. cube: V s 7 units pyramid: V Bh units solid: V 06 9 units. boxes, cylinders, prisms 5. cones, pyramids 6. a. True b. False 7. x yz xy 6yz xyz 8xy 6yz y 8. a. It is multiplied by, or 9. b. It is multiplied by, or oz gal, so 8 oz gal in. 8 oz oz in. x in. 8x x 5 8 The volume is about 5 cubic inches.. Given: ABC in sphere O is equilateral. To prove: moab mocb Argument: Conclusions Justifications 0. ABC in sphere O Given 0. is equilateral.. AB CB definition of equilateral triangle. OA OC definition of sphere. OB OB Reflexive Property of Equality. OBC OBA SSS Congruence Theorem 5. mocb moab CPCF Theorem. r r x x. r r r. Sample: Archimedes invented a screw now called the Archimedean screw for irrigating fields. He found formulas for the area of a segment of a parabola, and for the volumes of segments of paraboloids, hyperboloids, and spheroids. LESSON 0-9 pp There is no easy way to compute the area of its net, which is not -dimensional.. A sphere can be imagined as the union of almost pyramids whose base areas add to the surface area of the sphere.. Sphere Surface Area Formula: The total surface area S.A. of a sphere with radius r is r.. The surface area of a sphere is times the area of one of its great circles. 5. a. S.A. r 6 units b. 5 units 7

13 6. a. S.A. r 50 0,000 in. b. 0,000,6 in. 7. Earth: S.A ,76,00 97,060,797 square miles,600,000 square miles % 97,060,797 square miles The area of the United States is about.8% of the area of Earth. d 8. S.A. r d d 9. S.A ,5 5,0 square meters Cost $.05,0 $6,70 0. a. The moon has a radius about that of Earth. Since S.A. r, the radius is squared, so the moon has about, or, the surface 6 area of Earth. b. The moon has a radius about that of Earth. c. Since V r, the radius is cubed, so the moon has about, or, the volume of 6 Earth. 6. V r 6 r ; 6 7 r r S.A. r 6 m. sphere: S.A. r cylinder: L.A. ph rr r They are the same; both are r.. a. S.A square inches b. 6 The area of one cover must be about square inches.. V r cubic miles 5. V Bh h h Its height is units. 6. no base: sphere one base: pyramid, cone, regular pyramid, right cone two bases: prism, cylinder, right prism, box 7. Let r radius and h height of the tall jar. Tall jar: V Bh r h r h Short jar: V Bh r h r h The shorter jar holds twice as much jam. 8. a. L.A square inches b. Cylinder with height : Circumferencebase r 8.5 r 8.5 r 8.5 V Bh 6. cubic inches Cylinder with height 8.5: Circumferencebase r r V Bh r cubic inches The cylinder with height 8.5 has more volume. 9. First cube: V 97 units Second cube: V 597 7,65 units 7,65 65 units The length of an edge of the second cube is exactly 65 units. 7

14 0. a. AreaABCD x az a b. Volume x az ay a c. xz az ax a y a xyz ayz axy a y axz a z a x a. a. S.A cm b. One slab: S.A cm Number of slabs: The 0 cm edge is cut into 0 cm pieces, and one of the 5 cm edges is cut into 7.5 cm pieces. So there are 0 0 slabs. 060 cm 700 cm At least 700 cm of foil is needed to wrap all the slabs.. If a quadrilateral is a trapezoid, then it has at least two parallel sides.. Canada.0%, China.9%, Russia.%. a..7% b. 7.0% c. 66.8% C H A P T E R 0 PROGRESS SELF-TEST p. 66. a. Let x unknown leg length x 6 0 x x x L.A. ph units b. V Bh 6 0 = 880 units. L.A. ph 8 7h 6 h height 6 inches. V lwh 00 l l 8 l length 8 cm. a. L.A. lp units b. Let h height of the pyramid. h 0 6 h h 576 h V Bh 0 00 units 5. V Bh units 6. V Bh 96 B B 6 B The area of its base is 6 cm. 7. S.A. r 9 56 units V r ,7 units 8. S.A. r 00 r 5 r 5 r Its exact radius is 5 units L.A. lp lr rl. a. S.A. 69t 68t 86t units b. V 9t 79t units. The volume of the larger box is 7, or, times that of the smaller one. 7

15 . Jupiter s surface area is, or, times that of Earth.. a. Yes b. They have congruent bases and heights, and both volumes are 6h units. cylinder: V Bh 8 h 6h units prism: V Bh 6h 6h units 5. The volume of the pyramid is one-third the volume of the prism. 6. S.A m 7. L.A. lp in. 8. V r.5 59,7,000 miles 9. V Bh cm 0. x x 8 6x x x 8 6x 6x 8. V y x z 6 V xyz yz xz z 6xy 6y x C H A P T E R 0 REVIEW pp a. L.A. ph units b. S.A. L.A. B units c. V Bh 9 5 units. a. V Bh 0 90 units b. S.A. L.A. B ph B units. V Bh 5 70 units. V Bh 0 h 0 9h h 0. units 9 6. L.A. ph 60 6h 60 h h 5 V Bh cm 7. a. S.A. 5e 69e 6e, so length of edge is e units. b. V e 7e units 8. V Bh p p 6p p 6p units 9. Let l slant height of the cone. l units S.A. L.A. B lp B units V Bh units 0. V Bh ,000 units. a. slant height units b. L.A. lp units c. S.A. L.A. B ,00 units. V Bh units. L.A. lp units. V Bh 75 5 h 5 h height 5 cm 5. V s 5 s s 5 S.A units 75

16 5. V Bh 900 r 0 5 r r 5 5 r 6.6 ft 6. AD CD AC CD 6 CD 7 CD 600 CD 60 units V Bh units 7. 7, V s 50 s s The length of an edge of the cube is about 5. units S.A. r 7 0,76 65, units V r 7 97,66,56,58 units. S.A. r mm V r.5.5 mm. V r 88 r 6 r r 6 units. V r 0 r 0 r r 0 S.A. r 0. units 5. a. The surface area is multiplied by 9. b. The volume is multiplied by The volume is multiplied by A pizza is a cylinder, with V Bh r h. Since the radius is squared, the diameter is squared. The new volume is times as large. 8. The volume of the sun is 09,95,09,00,000 times the volume of Earth. 9. a. L.A. lp l s sl b. S.A. L.A. B sl s 0. a. L.A. ph r h rh b. V Bh r h r h. a. S.A. L.A. B lp B l r r rl r b. Let h height of the cone. h r l h l r h l r V Bh r l r. a. S.A. 6x 6x x units b. V x 8x units. No; Plane sections at any level other than the base do not have the same area.. a. True b. False 5. a. No b. Plane sections at any level do not have the same area. 76

17 6. a. No b. Plane sections at any level do not have the same area. 7. S.A in. 8. S.A. L.A. B ph B cm 9. Let l slant height l L.A. pl , cubits 0. L.A. pl cm. S.A. r 000 6,000, km. S.A. r in ft.9 ft You would need square feet of leather.. V in.. suitcase: V 6 ft 6 in. 0,68 in. bills: V,000, ,769 in. No, the suitcase cannot hold the bills since its volume is less than the volume of the bills. 5. V Bh m 6. V Bh cm 7. V Bh ,000 67,000 cubits 8. Let h height of the cone. h h 9 h 8 h 8 V Bh 8.0 in. 9. V r cm 50. V r 5.5,000 ft 5. 5x y 0xy 5x 8y 6 5. a 6a a 8 a a 6a 8 a a 6a 6a 0a 8 a 9a 0a 8 5. x 7x ; x x 8 5. r 7p q; rp 7p rq 7q 55. a 5b 9c 8; abc 8ab 9ac 7a 5bc 0b 5c x wy 5z 0; xyz wyz 5xz 5wz 0xy 0wy 50x 50w 77