# E t 8 8 15 sin 2 16 impossible 523 chapter 14 17 r

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: is as axis hyperbolic paraboloid straddling x- and z -axes paraboloid opening along the negative y -axis 6. (a) same (b) same (c) same 2 (d) same 7. (a) x = 0 : (e) y = 2 z x −2 2 a c (f ) y = z2 x2 z2 y2 + = 1; y = 0 : + = 1; 25 4 9 4 x2 z2 +2 a2 c z x2 z2 + =1 9 4 x2 y2 z=0: + =1 9 25 y2 z2 + =1 4 25 y x2 y2 + =1 9 25 x (b) x = 0 : z = 4y 2 ; y = 0 : z = x2 ; z z=0:x=y=0 z = 4 y2 z = x2 x2 + 4 y2 = 0 (0, 0, 0) x y Exercise Set 13.7 (c) x = 0 : 476 z2 x2 z2 y2 − = 1; y = 0 : − = 1; 16 4 9 4 z y2 z2 – =1 16 4 x2 y2 z=0: + =1 9 16 y y2 x2 + =1 9 16 x x2 z2 – =1 9 4 8. (a) x = 0 : y = z = 0; y = 0 : x = 9z 2 ; z = 0 : x = y 2 z x = 9z2 y x (b) x = 0 : −y 2 + 4z 2 = 4; y = 0 : x2 + z 2 = 1; x = y2 z=0 z z = 0 : 4x2 − y 2 = 4 y=0 x=0 y x y (c) x = 0 : z = ± ; y = 0 : z = ±x; z = 0 : x = y = 0 2 x=0 z z=0 x y y=0 9. (a) 4x2 + z 2 = 3; ellipse (b) y 2 + z 2 = 3; circle (c) y 2 + z 2 = 20; circle (d) 9x2 − y 2 = 20; hyperbola (e) z = 9x2 + 16; parabola (f ) 9x2 + 4y 2 = 4; ellipse 10. (a) y 2 − 4z 2 = 27; hyperbola (b) 9x2 + 4z 2 = 25; ellipse (c) 9z 2 − x2 = 4; hyperbola (d) x2 + 4y 2 = 9; ellipse (e) z = 1 − 4y 2 ; parabola (f ) x2 − 4y 2 = 4; hyperbola 477 Chapter 13 z 11. z 12. z 13. (0, 0, 3) (0, 0, 2) (1, 0, 0) x x y (0, 3, 0) (2, 0, 0) (0, 3, 0) (0, 2, 0) y x y (6, 0, 0) Ellipsoid Ellipsoid Hyperboloid of one sheet z 14. z 15. z 16. (0, 3, 0) (3, 0, 0) x y x Hyperboloid of one sheet (0, 0, 2) Elliptic cone Elliptic cone z 17. z 18. y x y z 19. (0, 0, –2) x y x y Hyperboloid of two sheets x y Hyperboloid of two sheets Hyperbolic paraboloid z 20. z 21. z 22. y x Hyperbolic paraboloid y x y x Elliptic paraboloid Circular paraboloid Exercise Set 13.7 478 z 23. z 24. z 25. (0, 0, 2) y x x y x (0, 2, 0) Elliptic paraboloid Hyperboloid of one sheet Circular cone z 26. z 27. y z 28. (0, 0, 2) (- 3, 0, 0) x (3, 0, 0) y Hyperboloid of two sheets (2, 0, 0) Hyperboloid of one sheet Hyperbolic paraboloid z 29. x y x 30. z (0, 0, 1) (0, 1, 0) y x y x z 31. (1, 0, 0) z 32. (0, 0, 1) x x (1, 0, 0) (0, 1, 0) y y y 479 Chapter 13 z 33. z 34. (0 , 0, 2 ) x y x y Hyperboloid of one sheet (–2, 3, –9) Circular paraboloid z 35. z 36. (-1 , 1 , 2 ) (1, –1, –2) y y x x Ellipsoid Hyperboloid of one sheet y2 x2 + =1 (b) 6, 4 9 4 (d) The focal axis is parallel to the x-axis. √ √ (c) (± 5, 0, 2) √ z2 y2 + =1 (b) 4, 2 2 4 2 (d) The focal axis is parallel to the y -axis. √ (c) (3, ± 2, 0) x2 y2 − =1 (b) (0, ±2, 4) 4 4 (d) The focal axis is parallel to the y -axis. √ (c) (0, ±2 2, 4) 37. (a) 38. (a) 39. (a) y2 x2 − =1 (b) (±2, 0, −4) 40. (a) 4 4 (e) The focal axis is parallel to the x-axis. 41. (a) z + 4 = y 2 (d) (c) (2, 0, −15/4) The focal axis is parallel to the z -axis. 42. (a) z − 4 = −x2 (d) (b) (2, 0, −4) √ (c) (±2 2, 0, −4) (b) (0, 2, 4) The focal axis is parallel to the z -axis. (c) (0, 2, 15/4) Exercise Set 13.7 480 43. x2 + y 2 = 4 − x2 − y 2 , x2 + y 2 = 2; circle of radius in the plane z = 2, centered at (0, 0, 1) √ 2 z 4 2 2 x +y = 2 (z = 2) x 44. y 2 + z = 4 − 2(y 2 + z ), y 2 + z = √ 3; 4/ parabolas in the planes x = ±2/ 3 which open in direction of the negative y -axis y z x y z = 4 – y2, x = 4 3 3 45. y = 4(x2 + z 2 ) 47. |z − (−1)| = paraboloid 46. y 2 = 4(x2 + z 2 ) x2 + y 2 + (z − 1)2 , z 2 + 2z + 1 = x2 + y 2 + z 2 − 2z + 1, z = (x2 + y 2 )/4; circular 48. |z + 1| = 2 x2 + y 2 + (z − 1)2 , z 2 + 2z + 1 = 4 x2 + y 2 + z 2 − 2z + 1 , 4x2 + 4y 2 + 3z 2 − 10z + 3 = 0, y2 (z − 5/3)2 x2 + + = 1; ellipsoid, center at (0, 0, 5/3). 4/3 4/3 16/9 x2 y2 x2 z2 + 2 = 1; if y = 0 then 2 + 2 = 1; since c < a the major axis has length 2a, the 2 a a a c minor axis length 2c. 49. If z = 0, 50. x2 y2 z2 + 2 + 2 = 1, where a = 6378.1370, b = 6356.5231. 2 a a b 51. Each slice perpendicular to the z -axis for |z | < c is an ellipse whose equation is y2 y2 c2 − z 2 x2 x2 +222 = 1, the area of which is + 2= , or 2 2 2 a2 b c2 (a /c )(c − z 2 ) (b /c )(c − z 2 ) π a c c2 − z 2 b c c2 − z 2 =π ab 2 c − z 2 so V = 2 c2 c π 0 ab 2 4 c − z 2 dz = πabc. 2 c 3 481 Chapter 13 EXERCISE SET 13.8 √ 5 2, 3π/4, 6 1. (a) (8, π/6, −4) (b) 2. (a) (2, 7π/4, 1) (b) (1, π/2, 1) (c) (2, π/2, 0) (d) (8, 5π/3, 6) √ (c) (4 2, 3π/4, −7) √ (d) (2 2, 7π/4, −2) (c) (5, 0, 4) (d) (−7, 0, −9) 3. (a) √ 2 3, 2, 3 (b) 4. (a) √ 3, −3 3, 7 (b) (0, 1, 0) (c) (0, 3, 5) (d) (0, 4, −1) 5. (a) √ 2 2, π/3, 3π/4 (b) (2, 7π/4, π/4) (c) (6, π/2, π/3) (d) (10, 5π/6, π/2) 6. (a) √ 8 2, π/4, π/6 (b) (c) (2, 0, π/2) (d) (4, π/6, π/6) √√ −4 2, 4 2, −2 √ 2 2, 5π/3, 3π/4 √ √ √ 7. (a) (5 6/4, 5 2/4, 5 2/2) (c) (0,0,1) (b) (7,0,0) (d) (0, −2, 0) √ √ √ − 2/4, 6/4, − 2/2 √√√ (c) (2 6, 2 2, 4 2) √ √ √ 3 2/4, −3 2/4, −3 3/2 √ (d) 0, 2 3, 2 8. (a) 9. (a) (b) √ 2 3, π/6, π/6 (c) (2, 3π/4, π/2) 10. (a) √ (b) 2, π/4, 3π/4 √ 4 3, 1, 2π/3 (d) √ 4 2, 5π/6, π/4 √ 2 2, 0, 3π/4 √...
View Full Document

## This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

Ask a homework question - tutors are online