Kindly solve 7.3 according sample solutionPlease provide me solution in 1 hour only 7.3 Heavy Metal(#differentiation)The temperature T in a metal ball is inversely proportional to the distance from thecenter of the ball, which we take to be the origin. The temperature at the point(1,3,2) is 100°C.(a) Derive a function T(x, y, z) that gives the temperature in the ball at the point(x, y, z).(b) Find the rate of change of T at (1,3,2) in the direction toward the point (0, 4, 2).(c) Show that at any point in the ball the direction of greatest increase in temper-ature is given by a vector that points toward the origin. Question 1. The temperature T in a metal ball is inversely proportional to the distance from thecenter of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 120°.(a) Find the rate of change of T at (1, 2, 2) in the direction toward the point (2, 1, 3).(b) Show that at any point in the ball the direction of greatest increase in temperature is given bya vector that points toward the origin.Solution 1. (a) The distance from (x, y, z) to the center is√x² + y² + 22so thatwhere k is a constant. Since T(1, 2, 2) = 120 weT(x, y, z) =A vector from (1, 2, 2) to (2, 1, 3) is (2, 1, 3)Computing the gradient, we findso thatT(x, y, z) =u=k√x² + y² + z2have that k = 360, thus360==/x² + y² + 2²(1,2,2)= (1,-1, 1). Normalizing it we find(1,-1,1).1√3VT(x, y, z) = -360(x² + y² + 2²)-(x, y, z),Du = VT u1= −360(x² + y² + 2²)−¹(x, y, z). (1,=(1,-1, 1)360√3Plugging in (1, 2, 2) we find DuT(1,2,2)= 3√340(b) The direction of greatest increase is parallel to the gradient of T. From (1) we see that thegradient points in the direction opposite to (x, y, z), hence toward the origin.(1)(x² + y² + ₂²)-³(x−y+z).

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7.3 Heavy Metal(#differentiation) The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1,3, 2) is 100°C. (a) Derive a function T(x, y, z) that gives the temperature in the ball at the point (x, y, z). (b) Find the rate of change of T at (1, 3, 2) in the direction toward the point (0, 4, 2). (c) Show that at any point in the ball the direction of greatest increase in temper- ature is given by a vector that points toward the origin.

7.3 Heavy Metal(#differentiation) The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1,3, 2) is 100°C. (a) Derive a function T(x, y, z) that gives the temperature in the ball at the point (x, y, z). (b) Find the rate of change of T at (1, 3, 2) in the direction toward the point (0, 4, 2). (c) Show that at any point in the ball the direction of greatest increase in temper- ature is given by a vector that points toward the origin.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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