Part I Given that the temperature distribution in our classroom can be described

Question

Part I

Given that the temperature distribution in our classroom can be described by

Part II

You are being chased by a mountain Yeti and in order to call for help you need to get to the top of the
mountain fast. The mountain

Given that the temperature distribution in our classroom can be described by T(x,y,z) = 50e -4x2 - 8y3 - 3z2 If your location within the classroom is (x,y,z)=(1,0.5,1) what direction does the temperature T have the maximum rate of change? In what direction does the temperature decrease the fastest? What is the maximum rate of change? You are being chased by a mountain Yeti and in order to call for help you need to get to the top of the mountain fast. The mountain's height is described by z = 2x3 + x2 - 5xy + 2000 If you are located at coordinate (x,y,z)=(10,10,3600), in what direction should you climb in order to reach the top of the mountain the fastest? There is a foul odor in the classroom that can be described by the odor function: F(x, y, z) = 500 + y2 - 15yz + 10xyz If you are sitting at coordinates (x,y,z)=(1,1,1), in what direction should you move in order to reduce the level of stench in the air to the greatest extent?

Explanation / Answer

part 1

dT/dx= 50 e-4x^2 -8y^3 -3z^2 * (-8x-0-0) = 50 e-4x^2 -8y^3 -3z^2 * (-8x)

dT/dy = 50 e-4x^2 -8y^3 -3z^2 * (0-24y-0) = 50 e-4x^2 -8y^3 -3z^2 * (-24y)

dT/dz = 50 e-4x^2 -8y^3 -3z^2 * (0-0-6z) = 50 e-4x^2 -8y^3 -3z^2 * (-6z)

At (1, 0.5, 1)

dT/dx = 50 e-4-1-3 * -8 = -400e-8

dT/dy = 50e-8 * -12 = -600 e-8

dT/dz = 50e-8 * -6 = -300e-8

maximum rate of change is in y direction

temperature decreases fastest in y direction

maximum rate of change is -600e^-8

part 2

dz/dx = 6x2 + 2x -5y

dz/dy = -5x

dz/dz =1

At (10,10,3600)

dz/dx= 6*100 + 2*10 -5*10 = 600+20-50 = 570

dz/dy = -50

dz/dz=1

climbing should be in x direction to reach fastest

part 3

dF/dx= 10yz

dF/dy = 2y -15z + 10 xz

dF/dz = -15y + 10xy

At (1,1,1)

dF/dx =10

dF/dy = 2-15+10 = -3

dF/dz= -15+10 = -5

The reduction in stench is maximum in z direction (-ve means reduction in stench)

pls rate :)

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