A metal plate is located in an xy plane such that the temperature T at (x; y) is
ID: 2831299 • Letter: A
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(x,y)=K*(1/(sqrt(x^2+y^2)))
dT/dt|x=K*-3/2*(1/(x^2+y^2)^-3/2*2x
dT/dt|y=K*-3/2*(1/(x^2+y^2)^-3/2*2y
at x=3 y=4
dT/dt|x=31.5
dT/dt|x=42
(-31i-42j)*K
B)in x direction
C) in y direction