Question

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 (1,2,2) is 120.(a) find the rate of change of T at (1,2,2) in the direction towards the point (2,1,3)(b) show that at any point in the ball the direction of greatest increase in temperature is given by the vector that points towards the origin.

Answer 1

this is wat i hv so far ... but nt getting the right answer.

Answer 2

this distance is called radius\[T=\frac{ k}{ r}\]
\[120=\frac{k}{r}\]
\[r=\sqrt{1^2+2^2+2^2}=3\]
\[k=120*3\]
so

Answer 3

\[T=\frac{ 360 }{r}\]

Answer 4

sorry I cant view the image

Answer 5

yes I have that but than how will i find the rate of change ? i applied the formula Du=del f (dot) unit vector
I posted the pic u can see wat i am taking abt

Answer 6

oh

Answer 7

Is this what you're getting for the gradient? Sorry it's really hard to read your paper since it's rotated....
\[\large T = \frac{ 360 }{ \sqrt{x^2+y^2+z^2} }\]\[\large \nabla T = \frac{ 360 }{ (x^2+y^2+z^2)^{3/2} }\left<x,y,z\right>\]

Answer 8

\[\huge \nabla T(1,2,2) = \frac{ 360 }{ \sqrt{1^2+2^2+2^2} }\left<1,2,2\right>\]

Answer 9

wait i am confused with ur formula

Answer 10

Yah sorry I simplified it quite a bit when i took the gradient. It looks really messy otherwise XD lol

Answer 11

\[\nabla T = \left<\frac{ 360(2x) }{ 2(x^2+y^2+z^2)^{3/2} },\frac{ 360(2y) }{ 2(x^2+y^2+z^2)^{3/2} },\frac{ 360(2z) }{ 2(x^2+y^2+z^2)^{3/2} }\right>\]

Answer 12

Taking the partials should give you this, I think. :O

Answer 13

oh i c where i was wrong i for the 2 in the denominator. gosh was trying to c where i was wrong since half an hour LOL
thanks i got it now !
can u help me with the end part of the question ?

Answer 14

hah XD

Answer 15

I had to do this question on a test a couple weeks back :D hmmm trying to remember the part b)...

Answer 16

oh really ? i hv test this week in 2 days

Answer 17

Hmmmm yah part b) I'm not really sure :c
Here is a brief solution I was able to find through googling... but I can't make sense of it XD lol
http://www.math.ucla.edu/~ronmiech/Calculus_Problems/32A/chap12/section6/811d25/811_25.html

Answer 18

yaa its not clear ..
any ways thanks of ur help :)