Answer to Question #269525 in Calculus for Megha

Question #269525

The temperature in the big hall is approximated by the function


T(x, y, z) = x


2 − 2xyz + z


2 + 5;


0 ≤ x ≤ 2, 0 ≤ y ≤ 3 and 0 ≤ z ≤ 2.


If a person located at (1, 1, 1), in which direction he should walk


to cool off as rapidly as possibly.


1
Expert's answer
2021-11-22T14:28:02-0500
"T(x, y, z)=x^2-2xyz+z^2+5"

"\\dfrac{\\partial T}{\\partial x}=2x-2yz"

"\\dfrac{\\partial T}{\\partial y}=-2xz"

"\\dfrac{\\partial T}{\\partial z}=-2xy+2z"

The gradient of "T" is


"\\nabla T=\\dfrac{\\partial T}{\\partial x}i+\\dfrac{\\partial T}{\\partial y}j+\\dfrac{\\partial T}{\\partial z}k"

"=(2x-2yz)i+(-2xz)j+(-2xy+2z)k"

At the point "(1,1,1)" the gradient vector is


"\\nabla T(1,1,1)=(2(1)-2(1)(1))i+(-2(1)(1))j"

"+(-2(1)(1)+2(1))k=-2j"

The temperature increases fastest in the direction of the gradient vector "\\nabla T(1,1,1)=-2j."

A person should walk in direction opposite to the gradient vector "\\nabla T(1,1,1)," or equivalently, in the direction of "2j" or the unit vector "j" to cool off as rapidly as possibly.

The maximum rate of decrease is the length of the gradient vector:


"|\\nabla T(1,1,1)|=|-2j|=2"

Therefore the maximum rate of decrease of temperature is "2\\ \\dfrac{unit\\ of\\ temperature}{unit \\ of\\ length}."



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS