Temperature is inversely proportional to position away from center

[clairaut]

A metal sphere with center at origin has its temperature that is inversely proportional to the position from center.

At point P(1,2,2) temperature is found to be 120 degrees

What is the rate of temperature change in direction of Q(2,1,3)?

 

[clairaut]

Here is my answer.

The center of the metal sphere has a high temperature larger than 120 degrees.

[T(x,y,z)] = -A (x^2 + y^2 + z^2) + C

Where A is the proportionality constant
And C is the temperature at center of metal sphere.

Gradient of
[T(1,2,2)] dot <1/rad3, -1/rad3, 1/rad3>
=
(-80 rad3)/9

Based upon an A = proportionality constant = -120/9 = -40/3
(This part is highly ambiguous to me but I just tried it for its quick simplicity)

My final answer is (-80 rad3)/9

The book says my answer is INCORRECT by a factor of 1/2

 

[LCKurtz]

A metal sphere with center at origin has its temperature that is inversely proportional to the position from center.

At point P(1,2,2) temperature is found to be 120 degrees

What is the rate of temperature change in direction of Q(2,1,3)?

Here is my answer.

The center of the metal sphere has a high temperature larger than 120 degrees.

[T(x,y,z)] = -A (x^2 + y^2 + z^2) + C

Have you stated the problem correctly? If the temperature is inversely proportional to the distance from the center you would start with$$
T = \frac k {\sqrt{x^2+y^2+z^2}}$$

 

[clairaut]

Oh... I did state the problem correctly.

I simply made the mistake of setting this equation up as a negative direct proportion.

Thank you.

I'll try it out again.