Chapter 15 — Answers

Selected answers for Chapter 15: Geometric Solids, transcribed from the book’s appendix. They are meant for checking your work, not for copying — the book prints answers to selected exercises only.

Back to Chapter 15 · Teacher’s answers: Chapter 15 · Complete key: Answers to Exercises PDF · Printed answers: appendices PDF

Lesson 1 (pages 620–624)

1. The second.

7. It appears to be oblique to the ground.

8. A vertical plane.

10. That they lie in a horizontal plane.

11. No. AB is perpendicular to the floor if it is perpendicular to every line in the floor that passes through point B.

13. Parallelograms.

15. EFGH.

21. They intersect in line l (or they both contain it).

23. One plane must contain a line that is perpendicular to the other plane.

25. CF is perpendicular to plane EFGH.

27. If a plane intersects two parallel planes, it intersects them in parallel lines.

32. Three noncollinear points determine a plane.

33. If two points lie in a plane, the line that contains them lies in the plane.

34. They lie in it.

40. No. Two horizontal lines can be skew. They can also intersect.

42. About 91 cm.

43. About 62°.

45. SAS.

47. A quadrilateral is a parallelogram if its diagonals bisect each other.

55. Nine.

Lesson 2 (pages 630–633)

1. Six.

2. Eight.

3. 12.

4. F.

7. CG, DH, EH, and FG.

10. 0.025 cm.

12. 0.225 cm².

14. 185 units.

15. 697 units.

18. $8x+10=106$.

20. A regular hexagon.

21. △HEM and △HEN are congruent right triangles.

25. ∠EAC is a right angle.

28. An isosceles trapezoid.

30. 6 cm.

32. 45°.

33. 60°.

35. About 35°.

39. $3\sqrt{2}$ cm.

44. $\dfrac{e}{2}\sqrt{3}$.

46. 54.74°.

49. One. The structure appears to contain $27+1=28$ cubes.

Lesson 3 (pages 635–638)

1. L, I, and E.

3. Prisms.

5. Two.

6. Three.

10. The bases.

12. $n$.

13. $n+2$.

18. 1,624 ft².

19. A parallelogram.

20. BAB′C.

23. EDE′F.

26. They appear to be parallel and equal.

29. Pentagons.

33. 1,458 ft².

35. They are rectangles.

38. Yes. The bases of every prism are congruent.

44. AX > AE.

46. A right triangular prism.

50. $2n$.

Lesson 4 (pages 642–646)

2. “x cubed.”

4. Its volume.

5. 27.

7. 36.

8. 1,296.

12. About 1.6 ft³.

14. 1,518,750 ft³.

16. 1 m³.

17. 6 m².

20. 12 m².

27. A right hexagonal prism and an oblique hexagonal prism.

31. 108 cubic units.

33. 288 cubic units.

34. 79 cm³.

36. 55 cm³.

38. 0.004 in.

40. 187.156 in².

41. $(a+b)^3$.

43. Two of them.

44. $a^2b$.

49. $ab+ac+bc+c\sqrt{a^2+b^2}$.

51. 3 cm.

54. 236.5 ft².

55. 4,730 ft³.

57. 9 ft³.

Lesson 5 (pages 649–653)

1. A pyramid.

2. Its apex.

4. A cross section.

5. A lateral face.

10. 128 cubic units.

12. 70 cubic units.

13. They are equal.

19. 576 yd³.

22. $V=\dfrac{1}{3}Bh$.

23. 55.125 square units.

25. 385.875 cubic units.

28. $\dfrac{1}{6}x^3$.

30. $\dfrac{1}{2}x$.

33. 231,000 ft².

34. 612 ft.

35. 231,000 ft².

37. 92,000,000 ft³.

38. $n+1$.

43. It is a pyramid.

45. Isosceles right triangles.

46. $\dfrac{3}{4}s^2$.

47. $\dfrac{1}{6}s^3$.

48. $\dfrac{1}{3}a^2(x+h)$.

51. OM = $\dfrac{1}{2}a$ and PN = $\dfrac{1}{2}b$.

52. Corresponding sides of similar triangles are proportional.

53. $\dfrac{x+h}{x}=\dfrac{a}{b}$.

Lesson 6 (pages 656–661)

6. Product.

10. Its axis.

11. The altitude.

12. The lateral surface.

13. The bases.

15. $45\pi$ in³.

19. 7 in².

22. 4,800 m³.

25. They have equal volumes.

26. Cavalieri’s Principle.

27. $35\pi$ cm³.

28. 110 cm³.

31. 65%.

34. That they have the same length.

38. The diameter of its base.

41. 352 in³.

42. $\pi h^2$.

44. It is multiplied by four.

48. 285 in³.

52. $2\pi rh$.

54. A sector of a circle.

55. $2\pi r$.

57. $\pi rl$.

p. 765

Lesson 7 (pages 665–669)

1. They are equal.

2. It is four times as large.

3. About 12.6 km².

4. About 4.2 km³.

7. $\dfrac{1}{3}\pi x^3$.

11. All radii of a sphere are equal.

12. If a line is perpendicular to a plane, it is perpendicular to every line in the plane that passes through the point of intersection.

18. It decreases.

19. 212 cm³.

21. 0.7 gram/cm³.

28. $\dfrac{2}{3}\pi r^3$.

30. $\dfrac{3}{2}\pi r^2$.

33. Either leg of a right triangle is the geometric mean between the hypotenuse and its projection on the hypotenuse.

35. 3,895 mi.

38. About 15 hours.

39. About 13 hours.

44. 325,000,000 mi³.

47. The second expression.

Lesson 8 (pages 672–676)

2. Two cubes are always similar.

6. It is multiplied by 9.

7. It is multiplied by 27.

10. It is multiplied by $n^2$.

11. It is multiplied by $n^3$.

13. 10.24.

14. 32.768.

17. That Gulliver’s body was similar to their bodies.

20. About 3.6 times.

23. About 22 mm.

25. About 1.8 times.

30. $24x^2$.

32. $8x^3$.

35. The square.

38. $x=\sqrt[3]{2}$; $y=\sqrt[3]{4}$.

39. PA, 1; PB, 2; PC, 4; PD, 8.

40. It is twice as large.

45. Eight.

51. About 47 pounds.

52. About 93 pounds.

Lesson 9 (pages 678–682)

1. Eight.

3. Quadrilateral.

6. Yes.

9. The area of the larger triangle is four times the area of the smaller one.

10. The surface area of the larger tetrahedron is $3^2=9$ times the surface area of the smaller one.

14. 12.

16. Three.

18. 20.

20. An icosahedron.

26. 12.

29. That they are congruent.

30. 180°.

34. 270°.

38. $\dfrac{e}{2}\sqrt{2}$.

41. $\dfrac{1}{6}e^3$.

44. ∠ABP = 54° and ∠BAP = 36°.

45. 1.618.

46. About 1.8.

52. 0.714 unit.

53. 10.5 square units.

58. 2.8 cubic units.

Review (pages 684–688)

1. 4 and 8.

3. They appear to be perpendicular.

10. 16 ft.

13. A prism and a pyramid.

15. 80,000 ft³.

18. It is four times as large.

21. 250,000.

23. 1,250,000,000.

25. $e\sqrt{2}$.

27. A regular tetrahedron.

29. $\dfrac{1}{6}e^3$.

30. $\dfrac{1}{3}e^3$.

36. $2\pi x^3$.

37. $\dfrac{2}{3}$.

40. $6\pi x^2$.

45. 40 square units.

50. The area of the horizontal cross section halfway up the barrel.

51. $\pi r^2h$.

58. Four.

59. Six.