(1 point The temperature at a point (x.y,z) is given by T(x, y, z) = 200e-z?/4-2
ID: 2895295 • Letter: #
Question
(1 point The temperature at a point (x.y,z) is given by T(x, y, z) = 200e-z?/4-22/9 measured in degrees celsius and x,y, and z in meters. There are lots of places to make silly errors in this problem; just try to keep track of what needs to be a unit vector. , where T is Find the rate of change of the temperature at the point (1,1, -1) in the direction toward the point (2, -4, -4). In which direction (unit vector) does the temperature increase the fastest at (1, 1, -1)? What is the maximum rate of increase of T at (1,1, -1)?Explanation / Answer
given T(x,y,z) =200ex2y2/4z2/9
gradient ,T=<200(-2x)ex2y2/4z2/9,200(-2y/4)ex2y2/4z2/9,200(-2z/9)ex2y2/4z2/9>
|T|=200ex2y2/4z2/9 [(-2x)2+(-2y/4)2+(-2z/9)2]
at (1,1,-1)
T=<-400e49/36,-100e49/36,(400/9)e49/36>
|T|=200e49/36[4+(1/4)+(4/81)]=(100/9)(1393)e49/36=(1001393)/(9e49/36)=106.32
a)
v=(2,-4,-4)-(1,1,-1) =<1,-5,-3>
|v|=[12+(-5)2+(-3)2] =35
rate of change of temperature at (1,1.-1) toward point (2,-4,-4) is=T.v/|v|
=<-400e49/36,-100e49/36,(400/9)e49/36>.<1,-5,-3>/35
=[-400+500+(-1200/9)]e49/36/35
=-(100/(335))e49/36
(b)
increasing fastest in direction T/|T|
=<-400e49/36,-100e49/36,(400/9)e49/36>/((100/9)(1393)e49/36)
=<-(36/1393),-(9/1393),(4/1393)>
(c)
maximum rate of increase =|T|=(1001393)/(9e49/36)=106.32
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