A metal plate is located in an xy plane such that the temperature T at (x; y) is

Question

A metal plate is located in an xy plane such that the temperature T at (x; y)
is inversely proportional to the distance from the origin, and the temperature at
P(3; 4) is 100 degrees F. (a) Find the rate of change of T at P in the direction i+j.
(b) In what direction does T Increase most rapidly at P ? (c) In what direction
does T decrease most rapidly at P?

Can someone help me on this calculus 3 problem, I know that Inversely proportional to the distance means that T = k*1/(distance) and that is going to make the equation be T(x,y)=K*(1/(sqrt(x^2+y^2))) but I dont know how to proceed. Thanks in advance :)

Explanation / Answer

T=K*(1/(sqrt(x^2+y^2))

Given at P(3,4) , T=100 degrees F

so substitute it above to get K

sqrt(9+16)=5;

So K = 500 ;

so T= 500/(sqrt(x^2+y^2)

now

a) x-direction=dt/dx = 500*(-1/2)*(x^2+y^2)^-3/2*(2x)=-500x(x^2+y^2)^-3/2

y direction.=dt/dy=similarly= -500y(x^2+y^2)^-3/2

b) at P(3,4)

dt/dx = -500x(x^2+y^2)^-3/2= -12

dt/dy=-500y(x^2+y^2)^-3/2=-16

so T decreases rapidly in both directions but more rapidly in y direction as shown

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