Mathematics, 16.03.2020 18:34 ebh3

The temperature at a point (x, y,z) is given by T(x, y,z)=200e^-x^2-3y^-9z^2, where T measured in degrees Celsius and x, y,z in meters. Find the rate of change of temperature at the point P(2, -1, 2) in the direction toward the point (3, -3, 3)

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Therefore the rate change of temperature at the point P(2,-1,2) in the direction toward the point (3,-3,3) is $-\frac{5200\sqrt6}{3}e^{-43}$ °C/m.

Step-by-step explanation:

Given that, the temperature at a point (x,y,z) is

$T(x,y,z)=200e^{-x^2-3y^2-9z^2}$ .

Rate change of temperature at the point P(2,-1,2) in the direction toward the point Q (3,-3,3) is D_uT(2,-1,2)

$T_x= 200(-2x)e^{-x^2-3y^2-9z^2}$ $=-400x200.2xe^{-x^2-3y^2-9z^2}$

$T_y = 200.(-6y)e^{-x^2-3y^2-9z^2}$ $=-1200ye^{-x^2-3y^2-9z^2}$

$T_z=200(-18z)e^{-x^2-3y^2-9z^2}$ $=-3600ze^{-x^2-3y^2-9z^2}$

The gradient of the temperature

$\bigtriangledown T(x,y,z)= (-400x200.2xe^{-x^2-3y^2-9z^2},-1200ye^{-x^2-3y^2-9z^2},-3600ze^{-x^2-3y^2-9z^2})$ $=-400e^{-x^2-3y^2-9z^2}(x,3y,9z)$

$\bigtriangledown T(2,-1,2) = -400e^{-2^2-3(-1)^2-9.2^2}(2,3.(-1),9.2)$

$=-400 e^{-43}(2,-3,18)$

V= $\overrightarrow {PQ}= \vec{Q}-\vec{P}$ =(3,-3,3)-(2,-1,2)=(1,-2,1)

The unit vector of V is $\frac{(1,-2,1)}{\sqrt{1^2+(-2)^2+1^2}}$

$=\frac{1}{\sqrt6}(1,-2,1)$

Therefore,

$D_uT(2,-1,2) = \bigtriangledown T(2,-1,2).\frac{1}{\sqrt6}(1,-2,1)$

$=-400 e^{-43}(2,-3,18).\frac{1}{\sqrt6}(1,-2,1)$

$=-\frac{400}{\sqrt6}e^{-43}(2.1+(-3).(-2)+18.1)$

$=-\frac{10400}{\sqrt6}e^{-43}$

$=-\frac{5200\sqrt6}{3}e^{-43}$ °C/m

Therefore the rate change of temperature at the point P(2,-1,2) in the direction toward the point (3,-3,3) is $-\frac{5200\sqrt6}{3}e^{-43}$ °C/m.

Answer from: Quest

Answer: d. 48 step-by-step explanation: i used a calculator so i am 100% sure these are right mark as brainliest?

Answer from: Quest

region b will make the inequalities y < 2x + 1 and y > -x - 2 true ⇒ answer a

step-by-step explanation:

* lets study the graph and the inequality to solve the question

- the inequality y > -x - 2

∵ the form of the inequality is y > mx + c , m is the slope of the line

and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = -2 that means the line intersect the y-axis at point (0 , -2)

∵ the sign of the inequality is >

∴ we shad the part over the line

region a and b

- the inequality y < 2x + 1

∵ the form of the inequality is y > mx + c , m is the slope of the line

and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = 1 that means the line intersect the y-axis at point (0 , 1)

∵ the sign of the inequality is <

∴ we shad the part under the line

region b and c

∵ the common region between the two lines is region b

∴ region b will make the inequalities y < 2x + 1 and y > -x - 2 true

Bwill make the inequalities y < 2x + 1 and y> -x- 2 true. o a. region b o b. region c o c. r

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The temperature at a point (x, y,z) is given by T(x, y,z)=200e^-x^2-3y^-9z^2, where T measured in de...

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