A cuboid-shaped room with dimensions shown in Figure Q2 is set up with a Cartesian coordinate...

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A cuboid-shaped room with dimensions shown in Figure Q2 is set up with a Cartesian coordinate system with the origin at one corner. The air temperature inside this room is described by the function T(x, y, z)=(x-1)²+(y-3)² +22³-62² +10 (a) (b) (c) (d) 5m xy 5m 3m (°C). Figure Q2 Calculate the gradient of the temperature function at (2, 1, 2). Hence or otherwise, find a unit vector in the direction of maximum temperature decrease at the point (2, 1, 2). (5 marks) A bee hovers at the point (2, 1, 2). If the bee starts to fly vertically downwards, will it experience a colder or hotter air temperature? What is the rate of temperature increase or decrease? (4 marks) Another bee is resting on the floor at (3, 2, 0). If it starts to fly towards the corner at (5, 5, 3), what is the rate of temperature increase or decrease the bee experiences? (6 marks) A vertical plane diagonally across the room is defined with corners at (0, 0, 0), (0, 0, 3), (5, 5, 0) and (5, 5, 3). Apply Lagrange multipliers to find the coldest and hottest locations on this plane inside the room. (10 marks) A cuboid-shaped room with dimensions shown in Figure Q2 is set up with a Cartesian coordinate system with the origin at one corner. The air temperature inside this room is described by the function T(x, y, z)=(x-1)²+(y-3)² +22³-62² +10 (a) (b) (c) (d) 5m xy 5m 3m (°C). Figure Q2 Calculate the gradient of the temperature function at (2, 1, 2). Hence or otherwise, find a unit vector in the direction of maximum temperature decrease at the point (2, 1, 2). (5 marks) A bee hovers at the point (2, 1, 2). If the bee starts to fly vertically downwards, will it experience a colder or hotter air temperature? What is the rate of temperature increase or decrease? (4 marks) Another bee is resting on the floor at (3, 2, 0). If it starts to fly towards the corner at (5, 5, 3), what is the rate of temperature increase or decrease the bee experiences? (6 marks) A vertical plane diagonally across the room is defined with corners at (0, 0, 0), (0, 0, 3), (5, 5, 0) and (5, 5, 3). Apply Lagrange multipliers to find the coldest and hottest locations on this plane inside the room. (10 marks)

Answer rating: 100% (QA)

a To calculate the gradient of the temperature function at 2 1 2 we need to find the partial derivatives with respect to each variable x y and z Tx 2x 1 Ty 1 Tz 2 Substituting the point 2 1 2 into the ... View the full answer