AN90 Answers: Chapter 11 27. y 5 x 5 3x 1 2y 5 6 2x 2 3y 5 0 29. y 5 x 6 x 2 2y 5 6 2x 2 4y 5 0 31. y 5 x 5 2x 1 y 5 2 2x 1 y 5 22 33. No solution y 5 x 5 2x 1 3y 5 0 2x 1 3y 5 6 35. y 5 x 5 x 1 y 5 3 x2 1 y2 5 9 37. y 5 x 5 y 5 x2 2 4 y 5 x 2 2 39. y 5 x 5 x2 1 y2 5 16 y 5 x2 2 4 41. y 5 x 5 xy 5 4 y 5 x2 1 1 43. Bounded y 8 x 8 x 1 2y 5 6 2x 1 y 5 6 (2, 2) (3, 0) (0, 0) (0, 3) 45. Unbounded y 5 x 5 2x 1 y 5 4 x 1 y 5 2 (0, 4) (2, 0) 47. Bounded y 8 x 8 2x 1 3y 5 12 3x 1 y 5 12 x 1 y 5 2 (0, 4) (0, 2) (2, 0) (4, 0) 12 7 , 24 7 49. Bounded y 5 x 9 2x 1 y 5 10 x 1 y 5 8 x 1 y 5 2 (2, 0) (5, 0) (2, 6) (0, 8) (0, 2) 51. Bounded y 16 x 16 x 1 2y 5 1 x 1 2y 5 10 (0, 5) (10, 0) (1, 0) 0, 1 2 53. ≤ + ≤ ≥ ≥ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ x x y x y 4 6 0 0 55. ≤ ≥ + ≤ − ≤ ≥ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ x y x y x y x 20 15 50 0 0 57. (a) + ≤ ≥ ≤ ≥ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ x y x y y 5000 3500 1000 0 (b) (3500, 1500) (3500, 1000) (3500, 0) (4000, 1000) y 2000 4000 x y 5 1000 x 5 3500 x 1 y 5 5000 (5000, 0) 59. (a) ≥ ≥ + ≤ + ≤ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ x y x y x y 0 0 2 300 3 2 480 (b) y 400 x 400 x 1 2y 5 300 3x 1 2y 5 480 (90, 105) (160, 0) (0, 150) (0, 0) 61. (a) + ≤ + ≤ ≥ ≥ ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ x y x y x y 3 2 160 2 3 150 0 0 (b) y 80 x 80 2x 1 3y 5 150 3x 1 2y 5 160 (0, 0) (0, 50) (36, 26) 160 3 , 0 63. { } − − − + i i 1 2 , 1 2 64. ( ) + − = x y 1 6 1 36 ; 2 2 circle with radius 1 6 and center ( ) 0, 1 6 ; 1 3 3P 4 U 5 5P 4 U 5 3P 2 U 5 7P 4 U 5 P 2 U 5 P 4 U 5 y x U 5 P U 5 0 1 6 65. ( ) ( ) − = − = f f 1 5; 2 28; So, there is a value x between −1 and 2 for which ( ) = f x 0. 66. π π { } 0, 2 3 , 4 3 67. { } [ ] − ≤ ≤ − x x 3 1 or 3,1 68. $8823.30 69. 18.75 horsepower 70. = x 5y 71. { } ≥− ≠ x x x 2, 23 72. Increasing: (−∞ − ⎤ ⎦ ⎥ ∞ , 1 3 , [5, ); Decreasing: ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 1 3 , 5 11.8 Assess Your Understanding (page 845) 1. objective function 2. T 3. Maximum value is 11; minimum value is 3. 5. Maximum value is 65; minimum value is 4. 7. Maximum value is 67; minimum value is 20. 9. The maximum value of z is 12, and it occurs at the point ( ) 6, 0 . 11. The minimum value of z is 4, and it occurs at the point ( ) 2, 0 . 13. The maximum value of z is 20, and it occurs at the point ( ) 0, 4 . 15. The minimum value of z is 8, and it occurs at the point ( ) 0, 2 . 17. The maximum value of z is 50, and it occurs at the point ( ) 10, 0 . 19. Produce 8 downhill and 24 cross-country; $1760; $1920 which is the profit when producing 16 downhill and 16 cross-country. 21. Rent 15 rectangular tables and 16 round tables for a minimum cost of $1252. 23. (a) $10,000 in a junk bond and $10,000 in Treasury bills (b) $12,000 in a junk bond and $8000 in Treasury bills 25. 120 lb of ground beef should be mixed with 60 lb of pork. 27. Manufacture 10 racing skates and 15 figure skates. 29. Order 2 metal samples and 4 plastic samples; $34 31. (a) Configure with 10 first class seats and 120 coach seats. (b) Configure with 15 first class seats and 120 coach seats. 34. { } − 1 32 , 1 35. y x 3 P 2P Domain: π { ≠ x x k , k is an integer}; range: ( ) −∞ ∞, 36. 89.1 years 37. = + y x3 7 38. 5 and 11 39. $13,303.89 40. + = x y 16 25 1 2 2 41. ( ) = − + − −f x x x 7 5 2 1 42. ( ) ( ) − − x x x 15 7 3 14 2 1 2 43. Concave up: ( )∞ 3, ; concave down: ( ) −∞, 3
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