contestada

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).

Respuesta :

Answer:

Rate of change [tex]24^{\circ}[/tex]C per meter.

Step-by-step explanation:

Given,

[tex]T(x,y,z)=300e-x_{2}-3y_{2}-7z_{2}[/tex]

Then,

[tex]T_{x_2}=-1, T_{y_2}=-3, T_{z_2}=-7[/tex]

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

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

[tex]=(6\times -1)+((-4)\times (-3))+(3\times 6)[/tex]

=[tex]24[/tex]

Hence the rate of change  of temperature is [tex]24^{\circ}[/tex]C per meter.

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