The temperature at a point (x,y,z) is given by T(x,y,z)=200e?x^2?y^2/4?z^2/9, wh
ID: 2851670 • Letter: T
Question
The temperature at a point (x,y,z) is given by T(x,y,z)=200e?x^2?y^2/4?z^2/9, where T is 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.
Find the rate of change of the temperature at the point (-1, 1, 2) in the direction toward the point (5, 4, -4).
In which direction (unit vector) does the temperature increase the fastest at (-1, 1, 2)?
What is the maximum rate of increase of T at (-1, 1, 2)?
The temperature at a point x y z is given by T x z = 20 e x-r/4 z 9 where T is 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. Find the rate of change of the temperature at the point (-1, 1, 2) in the direction toward the point (5, 4, -4). In which direction (unit vector) does the temperature increase the fastest at (-1, 1,2)? What is the maximum rate of increase of T at (1, 1, 2)?Explanation / Answer
T(x,y,z)=200ex^2 (y^2)/4 (z^2)/9
T=<-400xex^2 (y^2)/4 (z^2)/9,-100yex^2 (y^2)/4 (z^2)/9,(-400/9)zex^2 (y^2)/4 (z^2)/9>
T(-1,1,2)=<400e61/36,-100e61/36,(-800/9)e61/36>
p=(-1,1,2), q=(5,4,-4)
u=pq=q-p=(5,4,-4)-(-1,1,2)=<6,3,-6>
rate of change of temperature=directionderivative=T(-1,1,2).u/|u|
=<400e61/36,-100e61/36,(-800/9)e61/36>.<6,3,-6>/[62+(-3)2+62]
=<400e61/36,-100e61/36,(-800/9)e61/36>.<6,3,-6>/9
=<400e61/36,-100e61/36,(-800/9)e61/36>.<2,1,-2>/3
=(877.8/3)e61/36
rate of change of temperature=(877.8/3)e61/36=53.75
b)|T(-1,1,2)|=([4002+(-100)2+(-800/9)2])e61/36
unit vector =T(-1,1,2)/|T(-1,1,2)|
unit vector =<400e61/36,-100e61/36,(-800/9)e61/36>/([4002+(-100)2+(-800/9)2])e61/36
unit vector =<400,-100,(-800/9)>/([4002+(-100)2+(-800/9)2])=<0.948,-0.237,-0.211>
c)maximum inrease of T=|T(-1,1,2)| =([4002+(-100)2+(-800/9)2])e61/36=77.5
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