Question

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

Answer 1

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

Explanation:

Given equation of temperature is,

`T(x,y)=45-x^2-4y^2`

To find,

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

`\\abla T|_((2,-4))=(-2x, -8y)|_((2,-4))=(-4,32)`

which is the direction of point.

• Rate of increase is,

`\parallel \\abla T \parallel=√((-4)^2+(32)^2)=32.25` correct upto two desimal places.

Hence rate of temperature increase is 32.25 degree per centimeter.