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43. Find the points on the cone z² = x² + y² that are closest to the point (4, 2, 0).

43. Find the points on the cone z² = x² + y² that are closest to the point (4, 2, 0).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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E f(x, y) = 3xe' - x³ - e³y has exactly one critical point, and that f has a local maximum there that is not an absolute maximum. Then use a computer to produce a graph with a carefully chosen domain and view- point to see how this is possible. plane x + y 41. Find the shortest distance from the point (2, 0, -3) to the + z = 1. 42. Find the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 1, 1). 43. Find the points on the cone z² = x² + y² that are closest to the point (4, 2, 0). 44. Find the points on the surface y² = 9 + xz that are closest to the origin. 45. Find three positive numbers whose sum is 100 and whose product is a maximum. 46. Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible. 47. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. 48. Find the dimensions of the box with volume 1000 cm³ that has minimal surface area. 49. Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 6. 50. Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm². 01201327 im 51. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c. 55. If the length of the diagonal co what is the largest possible v 52. The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. 53. A cardboard box without a lid is to have a volume of 32,000 cm³. Find the dimensions that minimize the amount of cardboard used. ORTARE AL 54. A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m² per day, the north and south walls at a rate of floor at a rate of 1 unit/m² per day, 2 domain.) (c) Could you design a buildin the restrictions on the leng 56. A model for the yield Y of a of the nitrogen level N and I (measured in appropriate un Y(N, P) = where k is a positive consta phosphorus result in the be 57. The Shannon index (some index or Shannon-Weave an ecosystem. For the ca H=-Piln p where p, is the proportic (a) Express H as a func that pi + P2 + P3 (b) What is the domai (c) Find the maximur P1, P2, P3 does it c 58. Three alleles (alterna determine the four b BO), O (00), and A the proportion of in different alleles is P where p, q, and ra population. Use the at most 1/3. 59. Suppose that a sci quantities x and y at least approxima scientist performs form of points (x these points. The so the scientist w line y = mx + b figure). y YA
Transcribed Image Text:E f(x, y) = 3xe' - x³ - e³y has exactly one critical point, and that f has a local maximum there that is not an absolute maximum. Then use a computer to produce a graph with a carefully chosen domain and view- point to see how this is possible. plane x + y 41. Find the shortest distance from the point (2, 0, -3) to the + z = 1. 42. Find the point on the plane x - 2y + 3z = 6 that is closest to the point (0, 1, 1). 43. Find the points on the cone z² = x² + y² that are closest to the point (4, 2, 0). 44. Find the points on the surface y² = 9 + xz that are closest to the origin. 45. Find three positive numbers whose sum is 100 and whose product is a maximum. 46. Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible. 47. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r. 48. Find the dimensions of the box with volume 1000 cm³ that has minimal surface area. 49. Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 6. 50. Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm². 01201327 im 51. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c. 55. If the length of the diagonal co what is the largest possible v 52. The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. 53. A cardboard box without a lid is to have a volume of 32,000 cm³. Find the dimensions that minimize the amount of cardboard used. ORTARE AL 54. A rectangular building is being designed to minimize heat loss. The east and west walls lose heat at a rate of 10 units/m² per day, the north and south walls at a rate of floor at a rate of 1 unit/m² per day, 2 domain.) (c) Could you design a buildin the restrictions on the leng 56. A model for the yield Y of a of the nitrogen level N and I (measured in appropriate un Y(N, P) = where k is a positive consta phosphorus result in the be 57. The Shannon index (some index or Shannon-Weave an ecosystem. For the ca H=-Piln p where p, is the proportic (a) Express H as a func that pi + P2 + P3 (b) What is the domai (c) Find the maximur P1, P2, P3 does it c 58. Three alleles (alterna determine the four b BO), O (00), and A the proportion of in different alleles is P where p, q, and ra population. Use the at most 1/3. 59. Suppose that a sci quantities x and y at least approxima scientist performs form of points (x these points. The so the scientist w line y = mx + b figure). y YA
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