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The temperature in degrees Celsius on the surface of a metal plate is given by T(x, y), where x and y are measured in centimeters.

Find the direction from point P where the temperature increases most rapidly. ___________

T(x, y) = 45 − x^2 − 4y^2, P(2, −4)

Find the rate of increase. (Round your answer to two decimal places.)______________ ° per centimeter

Respuesta :

Answer:

Required direction is (-4,32) and rate of temperature increase is 32.25.

Step-by-step explanation:

Given equation of temperature is,

[tex]T(x,y)=45-x^2-4y^2[/tex]

To find,

  • Direction from point P(2,-4) where the temparature increase rapidly,

[tex]\nabla T|_{(2,-4)}=(-2x, -8y)|_{(2,-4)}=(-4,32)[/tex]

which is the direction of point.

  • Rate of increase is,

[tex]\parallel \nabla T \parallel=\sqrt{(-4)^2+(32)^2}=32.25[/tex]  correct upto two desimal places.

Hence rate of temperature increase is 32.25 degree per centimeter.

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