Suppose that the temperature T (x, y, z) of a spherical metal part satisfies, for each point (x, y, z) on its surface, which is inversely proportional to the distance from the point to the origin of coordinates (that is, T = k/D). Furthermore, it is known that The temperature at point A (1,2,2) is 120 ° C. a) Determine the value of the constant k. b) Determine the rate of change in temperature at point A, in the direction from A to point B (2,1,3). c) In which direction does the temperature increase faster in the point A? d) Verify that at any point (x, y, z) on the sphere, the direction towards which the temperature increases most rapidly is given by a vector pointing toward the coordinate origin.

Suppose that the temperature T (x, y, z) of a spherical metal part satisfies, for each point (x, y, z) on its surface, which is inversely proportional to the distance from the point to the origin of coordinates (that is, T = k/D). Furthermore, it is known that The temperature at point A (1,2,2) is 120 ° C. a) Determine the value of the constant k. b) Determine the rate of change in temperature at point A, in the direction from A to point B (2,1,3). c) In which direction does the temperature increase faster in the point A? d) Verify that at any point (x, y, z) on the sphere, the direction towards which the temperature increases most rapidly is given by a vector pointing toward the coordinate origin.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 67E

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