The temperature at a point (x,y,z) is given by T(xyz)=200e?x2?y24?z29 , where T

Question

The temperature at a point (x,y,z) is given by T(xyz)=200e?x2?y24?z29 , where T is measured in degrees celcius 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, -1) in the direction toward the point (5, 1, 1). 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

Unit vector towards (5,1,1) from (-1,-1,-1) = (6,2,2)/sqrt(44) = (6/6.63, 2/6.63, 2/6.63) ..... therefore, rate of change temperaure at (-1,-1,-1) towards (5,1,1) = 200e - 6/6.63 C .. Maximum temperature rate will chane in direction towards (1,1,1) .. :)

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