Answers to Exercises. Answers to Exercises 24.

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Answers to Exercises

28. False. This statement is true only for a right prism.

29. true 30. true

31. False. It is a sector of a circle. 32. true 33. false; counterexample: 34. true 35. true 2x x 2x x

CHAPTER 10 • CHAPTER CHAPTER 10 • CHAPTER

LESSON 10.1 1. polyhedron; polygonal; triangles 2.
PQR,
TUS 3. PQUT, QRSU, RPTS 4. QU

, PT
, RS
5. 6 cm 6. GYPTAN 7. point E 8. GE
, YE
, PE
, TE
, AE
, NE
9. 13 cm 10. D 11. L 12. C 13. G 14. B 15. H 16. E 17. A 18. J 19. J 20. M 21. H 22. I 23.

10

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Triangular Rectangular Pentagonal Hexagonal n-gonal

prism prism prism prism prism

Lateral 3 4 5 6 . . . n faces Total 5 6 7 8 . . . n
2 faces Edges 9 12 15 18 . . . 3n Vertices 6 8 10 12 . . . 2n 36. (Lesson 10.1) 36. See table below.

Possible answers include that the number of lateral faces of an antiprism is always twice the number for the related prism; that the number of vertices is the same for each related prism and antiprism; and that the number of edges for a prism is three times the number of faces, while for an antiprism, the number of edges is twice the number of faces. 37. Answer should include the idea that the painting “disappears” into the view out the window. Students might also note the effect created by the cone-shaped tower appearing similar to the road disappearing into the distance.

38. 8 39. 60 40. 30 41a. yes 41b. yes 41c. no 41d. yes ⫻ ⫻ ␣ ␣

Triangular Rectangular Pentagonal Hexagonal n-gonal

antiprism antiprism antiprism antiprism antiprism

Lateral 6 8 10 12 . . . 2n faces Total 8 10 12 14 . . . 2n
2 faces Edges 12 16 20 24 . . . 4n Vertices 6 8 10 12 . . . 2n

LESSON 10.2

1. 72 cm3 _{2. 24 cm}3 _{3. 108 cm}3

4. 160
cm3
502.65 cm3

5. 36
cm3
_{113.10 cm}3

6. 324
cm3
1017.88 cm3

7. See table below.

8. 960 in3 _{9. QT}
cubic units

10. sample answer:

11. 2x3 _{12. 3
r}3 _{13. 13x}3

14. Margaretta has room for 0.5625 cord. She should order a half cord.

15. 170 yd3 _{16. 5100 lb 17. 11,140}

18. The volume of the quilt in 1996 was 4000 ft3_.

The quilt panels were stacked 2 ft 8 in. high. 19. 2 12 12 12 8 3 Q T 24 in. 8 in. 12 in. 4 in. 20. 21. true 22. false 23.

24. possible solution: prism

25. approximately 1.89 m 26. 12
62 Salt crystal r r r

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Information about Height Right triangular Right rectangular Right trapezoidal Right

base of solid of solid prism prism prism cylinder

b 6, b₂ 7, H 20 a. V d. V g. V j. V h 8, r  3 480 cm3 _{960 cm}3 _{1040 cm}3 _180
cm3 b 9, b₂ 12, H 20 b. V e. V h. V k. V h 12, r  6 1080 cm3 _{2160 cm}3 _{2520 cm}3 ₇₂₀
cm3 b 8, b₂ 19, H 23 c. V f. V i. V l. V h 18, r  8 1656 cm3 _{3312 cm}3 _{5589 cm}3 _1472
cm3 H r b b ₂ H h H b h h b H 7. (Lesson 10.2)

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10. See table below. 11. V _31M2_{H ft}3

12. sample answer:

13. Mount Etna is larger. The volume for Mount Etna is approximately 2193 km3_{, and the volume}

for Mount Fuji is approximately 169 km3_.

48 3 27 16 H M M

Information about Height Triangular Rectangular Trapezoidal

base of solid of solid pyramid pyramid pyramid Cone

b 6, b₂ 7, H 20 a. V d. V g. V j. V h 6, r  3 b 9, b₂ 22, H 20 b. V e. V h. V k. V h 8, r  6 b 13, b₂ 29, H 24 c. V f. V i. V l. V h 17, r  8 H r b 2 b h H H b h h H b 120 cm3 240 cm3 884 cm3 240 cm3 480 cm3 1768 cm3 260 cm3 2856 cm3 60
cm3 240
cm3 512
cm3 24₃80cm3 14. 78,375 grams 15. 48 in3 16. 4
units3 17. 144x3_cm3 18. 40,200 gal; 44 h 40 min 19. 71 ft3 20. 403 barrels 21a. 16_3cm2 21b. 96 cm2 21c. 80_3cm2 21d. 2413  120 cm2 22.

Possible answer: From the properties of reflection, 1  3 and 2  4. m1  m2  90°, so m3  m4  90°, and m1  m2  m3  m4  180°. Therefore D, C, and D are collinear. 23a. Y (a c, d) 23b. Y (a c, b  d) 23c. Y (a c
e, b  d
f ) D' D'' C B A D 3 1 2 4 10. (Lesson 10.3)

LESSON 10.4 1. 58.5 in3 2. 32_{3 cm}3
55.43cm3 3. 15 cm 4. 11 cm 5. 5.0 cm 6.
8.₂5
in.2

The short, fat cylinder has greater volume. 7. 257 ft3

8. 4 cm

9. He must refute the statement. 10. 1502 lb

11. 192.4 gal 12. 13 min

13. Answers will vary, but
r2_{H should equal}

about 14.4 in3_.

14. approximately 38 in3

15. 100,000
m3_{, or about 314,159 m}3_;

16,528 loads

16. AB
 EC
because the opposite sides of a parallelogram are congruent. EC
 BD
because the diagonals of a rectangle are congruent. So, AB
 BD
because both are congruent to EC
. Therefore,
ABD is isosceles.

17. 8.2 cm 18. x 96° 19. 20. 21a. A 21b. S 21c. N 21d. S 21e. A x x x x x x A B C D 45° 45° 45° 45° 3x

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5. The gold has mass 2728.5 g, and the platinum has mass 7529.8 g. The solid cone of platinum has more mass.

6. 1.5 cm

7. 10.5 g/cm3_{; silver}

8. 8000 cm3

9. The volume of the medallion is 160 cm3_{. Yes, it}

is gold, and the Colonel is who he says he is. 10. 679 cm3 11. approximately 193 lb; 22 fish 12. ₂93 13. flowchart proof: 14. 15 sides 15a. (1, 3) 15b. (x
1)2 (y
3)2 25 16. 58; 3n
2






QR  SP







MQ  MP Given AIA ASA
RMQ 
SMP CPCTC R  S Vertical Angles QMR  PMS







SM  MR Given



M is the midpoint of PQ Definition of midpoint

LESSON 10.6 1. 36
cm3
_{113.1 cm}3 2. 
_6cm3
0.533 cm3 3. 9_3
2 cm3
_{0.884 cm}3 4. 720
cm3
2262 cm3 5. 30
cm3
_{94.3 cm}3 6. 3456
cm3
10,857 cm3 7. 18
m3
_{56.6 cm}3

8. No. The volume of the ice cream is 85.3

cm3_,

and the volume of the cone is 64
cm3_.

9. only 20 scoops

10. 14₃8 m
3_{, or about 155 m}3

11. They have the same volume. 12. 9 in.

13. 3 cm

14. 819₃ cm2
3
_{8579 cm}3

15. 18
in3_{, or about 57 in}3

16. No. The unused volume is 16
cm3_{, and the}

volume of the golf ball is 10.6

cm3_.

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17. approximately 15,704 gallons; 53 days 18. lithium

19. 31 ft

20. ABCD is a parallelogram because the slopes of CD

and AB
are both 0 and the slopes of BC
and AD

are both 5_3.

21. They trace two similar shapes, except that the one traced by C is smaller by a scale factor of 1: 2. 22. The line traces an infinite hourglass shape. Or, it traces the region between the two branches of a hyperbola. 23. w 110°, x  115°, y  80° y x D (9, 5) C (3, 5) A (3, –5) B (–3, –5)

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7. Area of great circle 
r2_{. Total surface area}

of hemisphere  3
r2_{. Total surface area of}

hemisphere is three times that of area of great circle. 8. 2 gal 9. V1_2

12. The total cost is $131.95. He will stay under budget. 13. 1.13% 14. 150
cm3
_{471.2 cm}3 15. 1_4 16. 1_2 17. 3_4

18. The ratio gets closer to 1. 19a and 19b. See tables below.

20. AB
 CB
and AD

 CD

by the definition of rhombus, and BD
 BD
because it is the same segment; therefore
ABD 
CBD by SSS. By CPCTC,2  3 and 1  4, which shows that BD
bisects both ABC and ADC. Because all four sides of the rhombus are congruent, a similar proof can be used to show that
ABC 
ADC and thus that A and C are both bisected by diagonal AC
.

21a. AB
 CB
by the definition of rhombus and BE
 BE
because it is the same segment. 1  2 by the Rhombus Angles Conjecture. Therefore,
AEB 
CEB by SAS.

21b. AE
 CE
by CPCTC, so BD
bisects AC
. 21c. 3  4 by CPCTC.Also, 3 and 4 form a linear pair, so they are supplementary. Because two angles that are congruent and supplementary are right angles,3 and 4 are right angles. 21d. Because 3 and 4 are right angles, the diagonals are perpendicular. You still need to show that AC
bisects BD
. Use a proof similar to that given in 21a to show that
AEB 
AED. Then, by CPCTC, BE
 DE
, which shows that AC
bisects BD
. n 1 2 3 4 5 6 . . . n . . . 200 f(n) 2 1 4 7 10 13 . . . 3n5 . . . 595 n 1 2 3 4 5 6 . . . n . . . 200 f(n) 0 1 3 12 35
23

57
. . .
n n
1 1
. . .
₂1₀99
₁ 19a. (Lesson 10.7) 19b. (Lesson 10.7)

USING YOUR ALGEBRA SKILLS 10 1. h
A_b
2. b
P
₂
or b 
P2h ₂

h 3. r



The original formula gives distance in terms of velocity and time.

8. F
9
C  32 ₅

The original formula converts degrees Fahrenheit to degrees Celsius.

9. L g





2

The original formula gives the period of a pendulum (time of one complete swing) in terms of length and acceleration due to gravity.

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11a. The corresponding radii are approximately 3.63 cm, 3.30 cm, 3.02 cm, and 2.79 cm.

11b. The corresponding heights are approximately 9.32 cm,10.49 cm,11.61 cm,and 12.70 cm.

11c. The corresponding volumes are approximately 129 cm3_{, 120 cm}3_{, 111 cm}3_{, and 104 cm}3_.

11d. Answers will vary. Sample answer: The cone with slant height 10 cm has the widest radius, so a scoop of ice cream is least likely to fall off, and that cone also has the greatest volume.

11e. Answers will vary. Sample answer: The cone with slant height 13 cm has the greatest height, so the cone appears bigger even though it has the same surface area as the other cones; that cone also has the smallest radius and smallest volume, so it could hold less ice cream and still appear to be a bigger cone.

12a. A

12b. average of 60: 31; average of 70: 56; average of 80: 81; average of 90: 106 (impossible)

m₁ m₂ m₃ f  f

 5

Pentahedron Hexahedron Octahedron Decahedron Dodecahedron

Number of faces

Number of edges 8 10 12 16 20

Number of vertices

5 6 8 10 12

5 6 6 8 10

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1. They have the same formula for volume: V BH. 2. They have the same formula for volume: V1_3BH. 3. 6240 cm3 4. 1029
cm3
3233 cm3 5. 1200 cm3 6. 32 cm3 7. 100
cm3
_{314.2 cm}3 8. 2250
cm3
7069 cm3 9. H 12.8 cm 10. h 7 cm 11. r 12 cm 12. r 8 cm 13. 960 cm3 14. 9 m 15. 851 cm3

16. four times as great 17a. V_{extra large} 54 in3

V_jumbo
201.1 in3

V_colossal
785.4 in3

17b. 14.5 times as great

18. Cylinder B weighs 8_3times as much as cylinder A. 19. 2129 kg; 9 loads 20. H2r.V V sp b h o e x re 
0.524. Thus, 52.4% of the box is filled by the ball.

21. approximately 358 yd3

22. No. The unused volume is 98
in3_{, and the}

volume of the meatballs is 32
in3_.

23. platinum

24. No. The ball weighs 253 lb. 25. 256 lb 26. approximately 3 in. 27.