Question

The temperature at a point (x, y, z) is given by T(x, y, z) = 300e−x2 − 3y2 − 7z2 where T is measured in °C and x, y, z in meters. Find the rate of change of temperature at the point P(2, −1, 3) in the direction towards the point (6, −4, 6).

Answer 1

Answer:

Rate of change $24^{\circ}$C per meter.

Step-by-step explanation:

Given,

$T(x,y,z)=300e-x_{2}-3y_{2}-7z_{2}$

Then,

$T_{x_2}=-1, T_{y_2}=-3, T_{z_2}=-7$

Change of temperature at the point say, $(x,y,z)=(2,-1,3)$ along the direction say, $(u,v,w)=(6,-4,6)$ is,

$\Big[uT_{x_2}+vT_{y_2}+wT_{z_2}\Big]_{(x,y,z)}$

$=(6\times -1)+((-4)\times (-3))+(3\times 6)$

=$24$

Hence the rate of change  of temperature is $24^{\circ}$C per meter.