Suppose that the temperature at a point inside a large ball of metal is inversel

Question

Suppose that the temperature at a point inside a large ball of metal is inversely proportional to the distance from the center of the ball, and that we choose a three-dimensional coordinate system in which the center of the ball is positioned at the origin.(a) If the temperature of the ball at the point (1, 2, 2) is 120 C, find the rate of change of the temperature at (1, 2, 2) in the direction of the point (2, 1, 3). (b) Prove that at any point inside the ball, the direction of greatest increase in temperature is given by a vector that points towards the center of the ball.

Explanation / Answer

a)

Temperature is inversely proportional to distance.

T=k/d=k/sqrt(x^2+y^2+z^2).

T= i T/x + j T/y + k T/z = - [1/(x^2+y^2+z^2)^(3/2)] * (ix+jy+kz)

T(1,2,2)=-1/27 * (i + 2j + 2k)

On the other hand, we know that the unit vector in (1,2,2) pointing towards (2,1,3) direction is ê=(1,-1,1)/sqrt(3).

Thus, T.ê = -1/27 * (i + 2j + 2k) . (i -j + k)/sqrt(3) = -1/(27xsqrt(3))

b) at a point (x,y,z), T= - [1/(x^2+y^2+z^2)^(3/2)] * (ix+jy+kz), the direction in which temperature has the greatest incresase will be the absolute value of T where it will be equal to Tmax= 1/(x^2+y^2+z^2)^2, which is the result of the dot product between T and the unit vector that points toward the center.

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