Calculus Multivariable McCallum, Hughes-Hallett, Gleason 4th Ed ch12 solutions

# Calculus: Multivariable

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12.1 SOLUTIONS 861 CHAPTER TWELVE Solutions for Section 12.1 Exercises 1. The distance of a point P = ( x, y, z ) from the yz -plane is | x | , from the xz -plane is | y | , and from the xy -plane is | z | . So, B is closest to the yz -plane, since it has the smallest x -coordinate in absolute value. B lies on the xz -plane, since its y -coordinate is 0 . B is farthest from the xy -plane, since it has the largest z -coordinate in absolute value. 2. The distance of a point P = ( x, y, z ) from the yz -plane is | x | , from the xz -plane is | y | , and from the xy -plane is | z | . So A is closest to the yz -plane, since it has the smallest x -coordinate in absolute value. B lies on the xz -plane, since its y -coordinate is 0 . C is farthest from the xy -plane, since it has the largest z -coordinate in absolute value. 3. Your final position is (1 , - 1 , - 3) . Therefore, you are in front of the yz -plane, to the left of the xz -plane, and below the xy -plane. 4. Your final position is (1 , - 1 , 1) . This places you in front of the yz -plane, to the left of the xz -plane, and above the xy -plane. 5. The point P is 1 2 + 2 2 + 1 2 = 6 = 2 . 45 units from the origin, and Q is 2 2 + 0 2 + 0 2 = 2 units from the origin. Since 2 < 6 , the point Q is closer. 6. The distance formula: d = p ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 gives us the distance between any pair of points ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 2 ) . Thus, we find Distance from P 1 to P 2 = 2 2 Distance from P 2 to P 3 = 6 Distance from P 1 to P 3 = 10 So P 2 and P 3 are closest to each other. 7. The equation is x 2 + y 2 + z 2 = 25 8. The equation is ( x - 1) 2 + ( y - 2) 2 + ( z - 3) 2 = 25 9. The graph is a plane parallel to the yz -plane, and passing through the point ( - 3 , 0 , 0) . See Figure 12.1. 31 x y z x = - 3 - 3 Figure 12.1 x y z y = 1 1 Figure 12.2 10. The graph is a plane parallel to the xz -plane, and passing through the point (0 , 1 , 0) . See Figure 12.2. 11. The graph is all points with y = 4 and z = 2 , i.e., a line parallel to the x -axis and passing through the points (0 , 4 , 2); (2 , 4 , 2); (4 , 4 , 2) etc. See Figure 12.3.

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862 Chapter Twelve /SOLUTIONS z x y 2 4 (0 , 4 , 2) (2 , 4 , 2) (4 , 4 , 2) Figure 12.3 12. (a) 80 - 90 F (b) 60 - 72 F (c) 60 - 100 F 13. - north south 100 90 80 70 Topeka distance from Topeka predicted high temperature Figure 12.4 14. North South 60 80 100 Boise Figure 12.5 West East 60 80 100 Boise Figure 12.6 15. The amount of money spent on beef equals the product of the unit price p and the quantity C of beef consumed: M = pC = pf ( I, p ) . Thus, we multiply each entry in Table 12.1 on page 605 of the text by the price at the top of the column. This yields Table 12.1. Table 12.1 Amount of money spent on beef ( \$ /household/week) Income Price 3 . 00 3 . 50 4 . 00 4 . 50 20 7 . 95 9 . 07 10 . 04 10 . 94 40 12 . 42 14 . 18 15 . 76 17 . 46 60 15 . 33 17 . 50 19 . 88 21 . 78 80 16 . 05 18 . 52 20 . 76 22 . 82 100 17 . 37 20 . 20 22 . 40 24 . 89
12.1 SOLUTIONS 863 16. Beef consumption by households making \$20 , 000 /year is given by Row 1 of Table 12.1 on page 605 of the text. Table 12.2 p 3 . 00 3 . 50 4 . 00 4 . 50 f (20 , p ) 2 . 65 2 . 59 2 . 51 2 . 43 For households making \$20 , 000 /year, beef consumption decreases as price goes up.

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